Question: What area of classification leads to troublesome arithmetic?

Answer: finite groups

Mathematicians often strive for a complete classification (or list) of a mathematical notion. <hl> In the context of finite groups , this aim leads to difficult mathematics. <hl> According to Lagrange's theorem, finite groups of order p, a prime number, are necessarily cyclic (abelian) groups Zp. Groups of order p2 can also be shown to be abelian, a statement which does not generalize to order p3, as the non-abelian group D4 of order 8 = 23 above shows. Computer algebra systems can be used to list small groups, but there is no classification of all finite groups.q[›] An intermediate step is the classification of finite simple groups.r[›] A nontrivial group is called simple if its only normal subgroups are the trivial group and the group itself.s[›] The Jordan–Hölder theorem exhibits finite simple groups as the building blocks for all finite groups. Listing all finite simple groups was a major achievement in contemporary group theory. 1998 Fields Medal winner Richard Borcherds succeeded in proving the monstrous moonshine conjectures, a surprising and deep relation between the largest finite simple sporadic group—the "monster group"—and certain modular functions, a piece of classical complex analysis, and string theory, a theory supposed to unify the description of many physical phenomena.

Mathematicians often strive for a complete classification (or list) of a mathematical notion. In the context of <hl> finite groups <hl>, this aim leads to difficult mathematics. According to Lagrange's theorem, finite groups of order p, a prime number, are necessarily cyclic (abelian) groups Zp. Groups of order p2 can also be shown to be abelian, a statement which does not generalize to order p3, as the non-abelian group D4 of order 8 = 23 above shows. Computer algebra systems can be used to list small groups, but there is no classification of all finite groups.q[›] An intermediate step is the classification of finite simple groups.r[›] A nontrivial group is called simple if its only normal subgroups are the trivial group and the group itself.s[›] The Jordan–Hölder theorem exhibits finite simple groups as the building blocks for all finite groups. Listing all finite simple groups was a major achievement in contemporary group theory. 1998 Fields Medal winner Richard Borcherds succeeded in proving the monstrous moonshine conjectures, a surprising and deep relation between the largest finite simple sporadic group—the "monster group"—and certain modular functions, a piece of classical complex analysis, and string theory, a theory supposed to unify the description of many physical phenomena.

In the context of <hl> finite groups <hl> , this aim leads to difficult mathematics.

question: What area of classification leads to troublesome arithmetic?, context: Mathematicians often strive for a complete classification (or list) of a mathematical notion. In the context of finite groups, this aim leads to difficult mathematics. According to Lagrange's theorem, finite groups of order p, a prime number, are necessarily cyclic (abelian) groups Zp. Groups of order p2 can also be shown to be abelian, a statement which does not generalize to order p3, as the non-abelian group D4 of order 8 = 23 above shows. Computer algebra systems can be used to list small groups, but there is no classification of all finite groups.q[›] An intermediate step is the classification of finite simple groups.r[›] A nontrivial group is called simple if its only normal subgroups are the trivial group and the group itself.s[›] The Jordan–Hölder theorem exhibits finite simple groups as the building blocks for all finite groups. Listing all finite simple groups was a major achievement in contemporary group theory. 1998 Fields Medal winner Richard Borcherds succeeded in proving the monstrous moonshine conjectures, a surprising and deep relation between the largest finite simple sporadic group—the "monster group"—and certain modular functions, a piece of classical complex analysis, and string theory, a theory supposed to unify the description of many physical phenomena.


Question: What defines finite groups of order p, a prime number, as being necessarily cyclic (abelian) groups Zp?

Answer: Lagrange's theorem

Mathematicians often strive for a complete classification (or list) of a mathematical notion. In the context of finite groups, this aim leads to difficult mathematics. <hl> According to Lagrange's theorem , finite groups of order p, a prime number, are necessarily cyclic (abelian) groups Zp. <hl> Groups of order p2 can also be shown to be abelian, a statement which does not generalize to order p3, as the non-abelian group D4 of order 8 = 23 above shows. Computer algebra systems can be used to list small groups, but there is no classification of all finite groups.q[›] An intermediate step is the classification of finite simple groups.r[›] A nontrivial group is called simple if its only normal subgroups are the trivial group and the group itself.s[›] The Jordan–Hölder theorem exhibits finite simple groups as the building blocks for all finite groups. Listing all finite simple groups was a major achievement in contemporary group theory. 1998 Fields Medal winner Richard Borcherds succeeded in proving the monstrous moonshine conjectures, a surprising and deep relation between the largest finite simple sporadic group—the "monster group"—and certain modular functions, a piece of classical complex analysis, and string theory, a theory supposed to unify the description of many physical phenomena.

Mathematicians often strive for a complete classification (or list) of a mathematical notion. In the context of finite groups, this aim leads to difficult mathematics. According to <hl> Lagrange's theorem <hl>, finite groups of order p, a prime number, are necessarily cyclic (abelian) groups Zp. Groups of order p2 can also be shown to be abelian, a statement which does not generalize to order p3, as the non-abelian group D4 of order 8 = 23 above shows. Computer algebra systems can be used to list small groups, but there is no classification of all finite groups.q[›] An intermediate step is the classification of finite simple groups.r[›] A nontrivial group is called simple if its only normal subgroups are the trivial group and the group itself.s[›] The Jordan–Hölder theorem exhibits finite simple groups as the building blocks for all finite groups. Listing all finite simple groups was a major achievement in contemporary group theory. 1998 Fields Medal winner Richard Borcherds succeeded in proving the monstrous moonshine conjectures, a surprising and deep relation between the largest finite simple sporadic group—the "monster group"—and certain modular functions, a piece of classical complex analysis, and string theory, a theory supposed to unify the description of many physical phenomena.

According to <hl> Lagrange's theorem <hl> , finite groups of order p, a prime number, are necessarily cyclic (abelian) groups Zp.

question: What defines finite groups of order p, a prime number, as being necessarily cyclic (abelian) groups Zp?, context: Mathematicians often strive for a complete classification (or list) of a mathematical notion. In the context of finite groups, this aim leads to difficult mathematics. According to Lagrange's theorem, finite groups of order p, a prime number, are necessarily cyclic (abelian) groups Zp. Groups of order p2 can also be shown to be abelian, a statement which does not generalize to order p3, as the non-abelian group D4 of order 8 = 23 above shows. Computer algebra systems can be used to list small groups, but there is no classification of all finite groups.q[›] An intermediate step is the classification of finite simple groups.r[›] A nontrivial group is called simple if its only normal subgroups are the trivial group and the group itself.s[›] The Jordan–Hölder theorem exhibits finite simple groups as the building blocks for all finite groups. Listing all finite simple groups was a major achievement in contemporary group theory. 1998 Fields Medal winner Richard Borcherds succeeded in proving the monstrous moonshine conjectures, a surprising and deep relation between the largest finite simple sporadic group—the "monster group"—and certain modular functions, a piece of classical complex analysis, and string theory, a theory supposed to unify the description of many physical phenomena.


Question: What can be used to classify small groups even though there is no classification of all finite groups?

Answer: Computer algebra systems

Mathematicians often strive for a complete classification (or list) of a mathematical notion. In the context of finite groups, this aim leads to difficult mathematics. According to Lagrange's theorem, finite groups of order p, a prime number, are necessarily cyclic (abelian) groups Zp. Groups of order p2 can also be shown to be abelian, a statement which does not generalize to order p3, as the non-abelian group D4 of order 8 = 23 above shows. <hl> Computer algebra systems can be used to list small groups, but there is no classification of all finite groups.q[›] An intermediate step is the classification of finite simple groups.r[›] A nontrivial group is called simple if its only normal subgroups are the trivial group and the group itself.s[›] The Jordan–Hölder theorem exhibits finite simple groups as the building blocks for all finite groups. <hl> Listing all finite simple groups was a major achievement in contemporary group theory. 1998 Fields Medal winner Richard Borcherds succeeded in proving the monstrous moonshine conjectures, a surprising and deep relation between the largest finite simple sporadic group—the "monster group"—and certain modular functions, a piece of classical complex analysis, and string theory, a theory supposed to unify the description of many physical phenomena.

Mathematicians often strive for a complete classification (or list) of a mathematical notion. In the context of finite groups, this aim leads to difficult mathematics. According to Lagrange's theorem, finite groups of order p, a prime number, are necessarily cyclic (abelian) groups Zp. Groups of order p2 can also be shown to be abelian, a statement which does not generalize to order p3, as the non-abelian group D4 of order 8 = 23 above shows. <hl> Computer algebra systems <hl> can be used to list small groups, but there is no classification of all finite groups.q[›] An intermediate step is the classification of finite simple groups.r[›] A nontrivial group is called simple if its only normal subgroups are the trivial group and the group itself.s[›] The Jordan–Hölder theorem exhibits finite simple groups as the building blocks for all finite groups. Listing all finite simple groups was a major achievement in contemporary group theory. 1998 Fields Medal winner Richard Borcherds succeeded in proving the monstrous moonshine conjectures, a surprising and deep relation between the largest finite simple sporadic group—the "monster group"—and certain modular functions, a piece of classical complex analysis, and string theory, a theory supposed to unify the description of many physical phenomena.

question: What can be used to classify small groups even though there is no classification of all finite groups?, context: Mathematicians often strive for a complete classification (or list) of a mathematical notion. In the context of finite groups, this aim leads to difficult mathematics. According to Lagrange's theorem, finite groups of order p, a prime number, are necessarily cyclic (abelian) groups Zp. Groups of order p2 can also be shown to be abelian, a statement which does not generalize to order p3, as the non-abelian group D4 of order 8 = 23 above shows. Computer algebra systems can be used to list small groups, but there is no classification of all finite groups.q[›] An intermediate step is the classification of finite simple groups.r[›] A nontrivial group is called simple if its only normal subgroups are the trivial group and the group itself.s[›] The Jordan–Hölder theorem exhibits finite simple groups as the building blocks for all finite groups. Listing all finite simple groups was a major achievement in contemporary group theory. 1998 Fields Medal winner Richard Borcherds succeeded in proving the monstrous moonshine conjectures, a surprising and deep relation between the largest finite simple sporadic group—the "monster group"—and certain modular functions, a piece of classical complex analysis, and string theory, a theory supposed to unify the description of many physical phenomena.


Question: What describes finite simple groups as the building pieces for all finite groups?

Answer: The Jordan–Hölder theorem

Mathematicians often strive for a complete classification (or list) of a mathematical notion. In the context of finite groups, this aim leads to difficult mathematics. According to Lagrange's theorem, finite groups of order p, a prime number, are necessarily cyclic (abelian) groups Zp. Groups of order p2 can also be shown to be abelian, a statement which does not generalize to order p3, as the non-abelian group D4 of order 8 = 23 above shows. <hl> Computer algebra systems can be used to list small groups, but there is no classification of all finite groups.q[›] An intermediate step is the classification of finite simple groups.r[›] A nontrivial group is called simple if its only normal subgroups are the trivial group and the group itself.s[›] The Jordan–Hölder theorem exhibits finite simple groups as the building blocks for all finite groups. <hl> Listing all finite simple groups was a major achievement in contemporary group theory. 1998 Fields Medal winner Richard Borcherds succeeded in proving the monstrous moonshine conjectures, a surprising and deep relation between the largest finite simple sporadic group—the "monster group"—and certain modular functions, a piece of classical complex analysis, and string theory, a theory supposed to unify the description of many physical phenomena.

Mathematicians often strive for a complete classification (or list) of a mathematical notion. In the context of finite groups, this aim leads to difficult mathematics. According to Lagrange's theorem, finite groups of order p, a prime number, are necessarily cyclic (abelian) groups Zp. Groups of order p2 can also be shown to be abelian, a statement which does not generalize to order p3, as the non-abelian group D4 of order 8 = 23 above shows. Computer algebra systems can be used to list small groups, but there is no classification of all finite groups.q[›] An intermediate step is the classification of finite simple groups.r[›] A nontrivial group is called simple if its only normal subgroups are the trivial group and the group itself.s[›] <hl> The Jordan–Hölder theorem <hl> exhibits finite simple groups as the building blocks for all finite groups. Listing all finite simple groups was a major achievement in contemporary group theory. 1998 Fields Medal winner Richard Borcherds succeeded in proving the monstrous moonshine conjectures, a surprising and deep relation between the largest finite simple sporadic group—the "monster group"—and certain modular functions, a piece of classical complex analysis, and string theory, a theory supposed to unify the description of many physical phenomena.

question: What describes finite simple groups as the building pieces for all finite groups?, context: Mathematicians often strive for a complete classification (or list) of a mathematical notion. In the context of finite groups, this aim leads to difficult mathematics. According to Lagrange's theorem, finite groups of order p, a prime number, are necessarily cyclic (abelian) groups Zp. Groups of order p2 can also be shown to be abelian, a statement which does not generalize to order p3, as the non-abelian group D4 of order 8 = 23 above shows. Computer algebra systems can be used to list small groups, but there is no classification of all finite groups.q[›] An intermediate step is the classification of finite simple groups.r[›] A nontrivial group is called simple if its only normal subgroups are the trivial group and the group itself.s[›] The Jordan–Hölder theorem exhibits finite simple groups as the building blocks for all finite groups. Listing all finite simple groups was a major achievement in contemporary group theory. 1998 Fields Medal winner Richard Borcherds succeeded in proving the monstrous moonshine conjectures, a surprising and deep relation between the largest finite simple sporadic group—the "monster group"—and certain modular functions, a piece of classical complex analysis, and string theory, a theory supposed to unify the description of many physical phenomena.


Question: What group operations must occur for group law and topology to integrate well?

Answer: continuous functions

Some topological spaces may be endowed with a group law. <hl> In order for the group law and the topology to interweave well, the group operations must be continuous functions , that is, g • h, and g−1 must not vary wildly if g and h vary only little. <hl> Such groups are called topological groups, and they are the group objects in the category of topological spaces. The most basic examples are the reals R under addition, (R ∖ {0}, ·), and similarly with any other topological field such as the complex numbers or p-adic numbers. All of these groups are locally compact, so they have Haar measures and can be studied via harmonic analysis. The former offer an abstract formalism of invariant integrals. Invariance means, in the case of real numbers for example:

Some topological spaces may be endowed with a group law. In order for the group law and the topology to interweave well, the group operations must be <hl> continuous functions <hl>, that is, g • h, and g−1 must not vary wildly if g and h vary only little. Such groups are called topological groups, and they are the group objects in the category of topological spaces. The most basic examples are the reals R under addition, (R ∖ {0}, ·), and similarly with any other topological field such as the complex numbers or p-adic numbers. All of these groups are locally compact, so they have Haar measures and can be studied via harmonic analysis. The former offer an abstract formalism of invariant integrals. Invariance means, in the case of real numbers for example:

In order for the group law and the topology to interweave well, the group operations must be <hl> continuous functions <hl> , that is, g • h, and g−1 must not vary wildly if g and h vary only little.

question: What group operations must occur for group law and topology to integrate well?, context: Some topological spaces may be endowed with a group law. In order for the group law and the topology to interweave well, the group operations must be continuous functions, that is, g • h, and g−1 must not vary wildly if g and h vary only little. Such groups are called topological groups, and they are the group objects in the category of topological spaces. The most basic examples are the reals R under addition, (R ∖ {0}, ·), and similarly with any other topological field such as the complex numbers or p-adic numbers. All of these groups are locally compact, so they have Haar measures and can be studied via harmonic analysis. The former offer an abstract formalism of invariant integrals. Invariance means, in the case of real numbers for example:


Question: What is the most simple example of topological groups?

Answer: reals R under addition, (R ∖ {0}, ·),

Some topological spaces may be endowed with a group law. In order for the group law and the topology to interweave well, the group operations must be continuous functions, that is, g • h, and g−1 must not vary wildly if g and h vary only little. Such groups are called topological groups, and they are the group objects in the category of topological spaces. <hl> The most basic examples are the reals R under addition, (R ∖ {0}, ·), and similarly with any other topological field such as the complex numbers or p-adic numbers. <hl> All of these groups are locally compact, so they have Haar measures and can be studied via harmonic analysis. The former offer an abstract formalism of invariant integrals. Invariance means, in the case of real numbers for example:

Some topological spaces may be endowed with a group law. In order for the group law and the topology to interweave well, the group operations must be continuous functions, that is, g • h, and g−1 must not vary wildly if g and h vary only little. Such groups are called topological groups, and they are the group objects in the category of topological spaces. The most basic examples are the <hl> reals R under addition, (R ∖ {0}, ·), <hl> and similarly with any other topological field such as the complex numbers or p-adic numbers. All of these groups are locally compact, so they have Haar measures and can be studied via harmonic analysis. The former offer an abstract formalism of invariant integrals. Invariance means, in the case of real numbers for example:

question: What is the most simple example of topological groups?, context: Some topological spaces may be endowed with a group law. In order for the group law and the topology to interweave well, the group operations must be continuous functions, that is, g • h, and g−1 must not vary wildly if g and h vary only little. Such groups are called topological groups, and they are the group objects in the category of topological spaces. The most basic examples are the reals R under addition, (R ∖ {0}, ·), and similarly with any other topological field such as the complex numbers or p-adic numbers. All of these groups are locally compact, so they have Haar measures and can be studied via harmonic analysis. The former offer an abstract formalism of invariant integrals. Invariance means, in the case of real numbers for example:


Question: What variables do locally compact groups share that can be studied by harmonic analysis?

Answer: Haar measures

Some topological spaces may be endowed with a group law. In order for the group law and the topology to interweave well, the group operations must be continuous functions, that is, g • h, and g−1 must not vary wildly if g and h vary only little. Such groups are called topological groups, and they are the group objects in the category of topological spaces. The most basic examples are the reals R under addition, (R ∖ {0}, ·), and similarly with any other topological field such as the complex numbers or p-adic numbers. <hl> All of these groups are locally compact, so they have Haar measures and can be studied via harmonic analysis. <hl> The former offer an abstract formalism of invariant integrals. Invariance means, in the case of real numbers for example:

Some topological spaces may be endowed with a group law. In order for the group law and the topology to interweave well, the group operations must be continuous functions, that is, g • h, and g−1 must not vary wildly if g and h vary only little. Such groups are called topological groups, and they are the group objects in the category of topological spaces. The most basic examples are the reals R under addition, (R ∖ {0}, ·), and similarly with any other topological field such as the complex numbers or p-adic numbers. All of these groups are locally compact, so they have <hl> Haar measures <hl> and can be studied via harmonic analysis. The former offer an abstract formalism of invariant integrals. Invariance means, in the case of real numbers for example:

question: What variables do locally compact groups share that can be studied by harmonic analysis?, context: Some topological spaces may be endowed with a group law. In order for the group law and the topology to interweave well, the group operations must be continuous functions, that is, g • h, and g−1 must not vary wildly if g and h vary only little. Such groups are called topological groups, and they are the group objects in the category of topological spaces. The most basic examples are the reals R under addition, (R ∖ {0}, ·), and similarly with any other topological field such as the complex numbers or p-adic numbers. All of these groups are locally compact, so they have Haar measures and can be studied via harmonic analysis. The former offer an abstract formalism of invariant integrals. Invariance means, in the case of real numbers for example:


Question: What concepts are fundamental to number theory?

Answer: adele rings and adelic algebraic groups

<hl> for any constant c. Matrix groups over these fields fall under this regime, as do adele rings and adelic algebraic groups , which are basic to number theory. <hl> Galois groups of infinite field extensions such as the absolute Galois group can also be equipped with a topology, the so-called Krull topology, which in turn is central to generalize the above sketched connection of fields and groups to infinite field extensions. An advanced generalization of this idea, adapted to the needs of algebraic geometry, is the étale fundamental group.

for any constant c. Matrix groups over these fields fall under this regime, as do <hl> adele rings and adelic algebraic groups <hl>, which are basic to number theory. Galois groups of infinite field extensions such as the absolute Galois group can also be equipped with a topology, the so-called Krull topology, which in turn is central to generalize the above sketched connection of fields and groups to infinite field extensions. An advanced generalization of this idea, adapted to the needs of algebraic geometry, is the étale fundamental group.

for any constant c. Matrix groups over these fields fall under this regime, as do <hl> adele rings and adelic algebraic groups <hl> , which are basic to number theory.

question: What concepts are fundamental to number theory?, context: for any constant c. Matrix groups over these fields fall under this regime, as do adele rings and adelic algebraic groups, which are basic to number theory. Galois groups of infinite field extensions such as the absolute Galois group can also be equipped with a topology, the so-called Krull topology, which in turn is central to generalize the above sketched connection of fields and groups to infinite field extensions. An advanced generalization of this idea, adapted to the needs of algebraic geometry, is the étale fundamental group.


Question: What group uses infinite field extensions with topology?

Answer: the absolute Galois group

for any constant c. Matrix groups over these fields fall under this regime, as do adele rings and adelic algebraic groups, which are basic to number theory. <hl> Galois groups of infinite field extensions such as the absolute Galois group can also be equipped with a topology, the so-called Krull topology, which in turn is central to generalize the above sketched connection of fields and groups to infinite field extensions. <hl> An advanced generalization of this idea, adapted to the needs of algebraic geometry, is the étale fundamental group.

for any constant c. Matrix groups over these fields fall under this regime, as do adele rings and adelic algebraic groups, which are basic to number theory. Galois groups of infinite field extensions such as <hl> the absolute Galois group <hl> can also be equipped with a topology, the so-called Krull topology, which in turn is central to generalize the above sketched connection of fields and groups to infinite field extensions. An advanced generalization of this idea, adapted to the needs of algebraic geometry, is the étale fundamental group.

question: What group uses infinite field extensions with topology?, context: for any constant c. Matrix groups over these fields fall under this regime, as do adele rings and adelic algebraic groups, which are basic to number theory. Galois groups of infinite field extensions such as the absolute Galois group can also be equipped with a topology, the so-called Krull topology, which in turn is central to generalize the above sketched connection of fields and groups to infinite field extensions. An advanced generalization of this idea, adapted to the needs of algebraic geometry, is the étale fundamental group.


Question: What is used to generalize the connection of fields and groups to infinite field extensions?

Answer: Krull topology

for any constant c. Matrix groups over these fields fall under this regime, as do adele rings and adelic algebraic groups, which are basic to number theory. <hl> Galois groups of infinite field extensions such as the absolute Galois group can also be equipped with a topology, the so-called Krull topology , which in turn is central to generalize the above sketched connection of fields and groups to infinite field extensions. <hl> An advanced generalization of this idea, adapted to the needs of algebraic geometry, is the étale fundamental group.

for any constant c. Matrix groups over these fields fall under this regime, as do adele rings and adelic algebraic groups, which are basic to number theory. Galois groups of infinite field extensions such as the absolute Galois group can also be equipped with a topology, the so-called <hl> Krull topology <hl>, which in turn is central to generalize the above sketched connection of fields and groups to infinite field extensions. An advanced generalization of this idea, adapted to the needs of algebraic geometry, is the étale fundamental group.

Galois groups of infinite field extensions such as the absolute Galois group can also be equipped with a topology, the so-called <hl> Krull topology <hl> , which in turn is central to generalize the above sketched connection of fields and groups to infinite field extensions.

question: What is used to generalize the connection of fields and groups to infinite field extensions?, context: for any constant c. Matrix groups over these fields fall under this regime, as do adele rings and adelic algebraic groups, which are basic to number theory. Galois groups of infinite field extensions such as the absolute Galois group can also be equipped with a topology, the so-called Krull topology, which in turn is central to generalize the above sketched connection of fields and groups to infinite field extensions. An advanced generalization of this idea, adapted to the needs of algebraic geometry, is the étale fundamental group.


Question: What group is an advanced observation of infinite field extensions and groups that is adapted for the needs of algebraic geometry?

Answer: the étale fundamental group

for any constant c. Matrix groups over these fields fall under this regime, as do adele rings and adelic algebraic groups, which are basic to number theory. Galois groups of infinite field extensions such as the absolute Galois group can also be equipped with a topology, the so-called Krull topology, which in turn is central to generalize the above sketched connection of fields and groups to infinite field extensions. <hl> An advanced generalization of this idea, adapted to the needs of algebraic geometry, is the étale fundamental group . <hl>

for any constant c. Matrix groups over these fields fall under this regime, as do adele rings and adelic algebraic groups, which are basic to number theory. Galois groups of infinite field extensions such as the absolute Galois group can also be equipped with a topology, the so-called Krull topology, which in turn is central to generalize the above sketched connection of fields and groups to infinite field extensions. An advanced generalization of this idea, adapted to the needs of algebraic geometry, is <hl> the étale fundamental group <hl>.

An advanced generalization of this idea, adapted to the needs of algebraic geometry, is <hl> the étale fundamental group <hl> .

question: What group is an advanced observation of infinite field extensions and groups that is adapted for the needs of algebraic geometry?, context: for any constant c. Matrix groups over these fields fall under this regime, as do adele rings and adelic algebraic groups, which are basic to number theory. Galois groups of infinite field extensions such as the absolute Galois group can also be equipped with a topology, the so-called Krull topology, which in turn is central to generalize the above sketched connection of fields and groups to infinite field extensions. An advanced generalization of this idea, adapted to the needs of algebraic geometry, is the étale fundamental group.


Question: What concept is of basic importance in modern physics?

Answer: Lie groups

<hl> Lie groups are of fundamental importance in modern physics: <hl> Noether's theorem links continuous symmetries to conserved quantities. Rotation, as well as translations in space and time are basic symmetries of the laws of mechanics. They can, for instance, be used to construct simple models—imposing, say, axial symmetry on a situation will typically lead to significant simplification in the equations one needs to solve to provide a physical description.v[›] Another example are the Lorentz transformations, which relate measurements of time and velocity of two observers in motion relative to each other. They can be deduced in a purely group-theoretical way, by expressing the transformations as a rotational symmetry of Minkowski space. The latter serves—in the absence of significant gravitation—as a model of space time in special relativity. The full symmetry group of Minkowski space, i.e. including translations, is known as the Poincaré group. By the above, it plays a pivotal role in special relativity and, by implication, for quantum field theories. Symmetries that vary with location are central to the modern description of physical interactions with the help of gauge theory.

<hl> Lie groups <hl> are of fundamental importance in modern physics: Noether's theorem links continuous symmetries to conserved quantities. Rotation, as well as translations in space and time are basic symmetries of the laws of mechanics. They can, for instance, be used to construct simple models—imposing, say, axial symmetry on a situation will typically lead to significant simplification in the equations one needs to solve to provide a physical description.v[›] Another example are the Lorentz transformations, which relate measurements of time and velocity of two observers in motion relative to each other. They can be deduced in a purely group-theoretical way, by expressing the transformations as a rotational symmetry of Minkowski space. The latter serves—in the absence of significant gravitation—as a model of space time in special relativity. The full symmetry group of Minkowski space, i.e. including translations, is known as the Poincaré group. By the above, it plays a pivotal role in special relativity and, by implication, for quantum field theories. Symmetries that vary with location are central to the modern description of physical interactions with the help of gauge theory.

question: What concept is of basic importance in modern physics?, context: Lie groups are of fundamental importance in modern physics: Noether's theorem links continuous symmetries to conserved quantities. Rotation, as well as translations in space and time are basic symmetries of the laws of mechanics. They can, for instance, be used to construct simple models—imposing, say, axial symmetry on a situation will typically lead to significant simplification in the equations one needs to solve to provide a physical description.v[›] Another example are the Lorentz transformations, which relate measurements of time and velocity of two observers in motion relative to each other. They can be deduced in a purely group-theoretical way, by expressing the transformations as a rotational symmetry of Minkowski space. The latter serves—in the absence of significant gravitation—as a model of space time in special relativity. The full symmetry group of Minkowski space, i.e. including translations, is known as the Poincaré group. By the above, it plays a pivotal role in special relativity and, by implication, for quantum field theories. Symmetries that vary with location are central to the modern description of physical interactions with the help of gauge theory.


Question: What connects continuous symmetries to conserved quantities?

Answer: Noether's theorem

<hl> Lie groups are of fundamental importance in modern physics: Noether's theorem links continuous symmetries to conserved quantities. <hl> Rotation, as well as translations in space and time are basic symmetries of the laws of mechanics. They can, for instance, be used to construct simple models—imposing, say, axial symmetry on a situation will typically lead to significant simplification in the equations one needs to solve to provide a physical description.v[›] Another example are the Lorentz transformations, which relate measurements of time and velocity of two observers in motion relative to each other. They can be deduced in a purely group-theoretical way, by expressing the transformations as a rotational symmetry of Minkowski space. The latter serves—in the absence of significant gravitation—as a model of space time in special relativity. The full symmetry group of Minkowski space, i.e. including translations, is known as the Poincaré group. By the above, it plays a pivotal role in special relativity and, by implication, for quantum field theories. Symmetries that vary with location are central to the modern description of physical interactions with the help of gauge theory.

Lie groups are of fundamental importance in modern physics: <hl> Noether's theorem <hl> links continuous symmetries to conserved quantities. Rotation, as well as translations in space and time are basic symmetries of the laws of mechanics. They can, for instance, be used to construct simple models—imposing, say, axial symmetry on a situation will typically lead to significant simplification in the equations one needs to solve to provide a physical description.v[›] Another example are the Lorentz transformations, which relate measurements of time and velocity of two observers in motion relative to each other. They can be deduced in a purely group-theoretical way, by expressing the transformations as a rotational symmetry of Minkowski space. The latter serves—in the absence of significant gravitation—as a model of space time in special relativity. The full symmetry group of Minkowski space, i.e. including translations, is known as the Poincaré group. By the above, it plays a pivotal role in special relativity and, by implication, for quantum field theories. Symmetries that vary with location are central to the modern description of physical interactions with the help of gauge theory.

question: What connects continuous symmetries to conserved quantities?, context: Lie groups are of fundamental importance in modern physics: Noether's theorem links continuous symmetries to conserved quantities. Rotation, as well as translations in space and time are basic symmetries of the laws of mechanics. They can, for instance, be used to construct simple models—imposing, say, axial symmetry on a situation will typically lead to significant simplification in the equations one needs to solve to provide a physical description.v[›] Another example are the Lorentz transformations, which relate measurements of time and velocity of two observers in motion relative to each other. They can be deduced in a purely group-theoretical way, by expressing the transformations as a rotational symmetry of Minkowski space. The latter serves—in the absence of significant gravitation—as a model of space time in special relativity. The full symmetry group of Minkowski space, i.e. including translations, is known as the Poincaré group. By the above, it plays a pivotal role in special relativity and, by implication, for quantum field theories. Symmetries that vary with location are central to the modern description of physical interactions with the help of gauge theory.


Question: What term describes the basic symmetries of the laws of mechanics?

Answer: Rotation

Lie groups are of fundamental importance in modern physics: Noether's theorem links continuous symmetries to conserved quantities. <hl> Rotation , as well as translations in space and time are basic symmetries of the laws of mechanics. <hl> They can, for instance, be used to construct simple models—imposing, say, axial symmetry on a situation will typically lead to significant simplification in the equations one needs to solve to provide a physical description.v[›] Another example are the Lorentz transformations, which relate measurements of time and velocity of two observers in motion relative to each other. They can be deduced in a purely group-theoretical way, by expressing the transformations as a rotational symmetry of Minkowski space. The latter serves—in the absence of significant gravitation—as a model of space time in special relativity. The full symmetry group of Minkowski space, i.e. including translations, is known as the Poincaré group. By the above, it plays a pivotal role in special relativity and, by implication, for quantum field theories. Symmetries that vary with location are central to the modern description of physical interactions with the help of gauge theory.

Lie groups are of fundamental importance in modern physics: Noether's theorem links continuous symmetries to conserved quantities. <hl> Rotation <hl>, as well as translations in space and time are basic symmetries of the laws of mechanics. They can, for instance, be used to construct simple models—imposing, say, axial symmetry on a situation will typically lead to significant simplification in the equations one needs to solve to provide a physical description.v[›] Another example are the Lorentz transformations, which relate measurements of time and velocity of two observers in motion relative to each other. They can be deduced in a purely group-theoretical way, by expressing the transformations as a rotational symmetry of Minkowski space. The latter serves—in the absence of significant gravitation—as a model of space time in special relativity. The full symmetry group of Minkowski space, i.e. including translations, is known as the Poincaré group. By the above, it plays a pivotal role in special relativity and, by implication, for quantum field theories. Symmetries that vary with location are central to the modern description of physical interactions with the help of gauge theory.

<hl> Rotation <hl> , as well as translations in space and time are basic symmetries of the laws of mechanics.

question: What term describes the basic symmetries of the laws of mechanics?, context: Lie groups are of fundamental importance in modern physics: Noether's theorem links continuous symmetries to conserved quantities. Rotation, as well as translations in space and time are basic symmetries of the laws of mechanics. They can, for instance, be used to construct simple models—imposing, say, axial symmetry on a situation will typically lead to significant simplification in the equations one needs to solve to provide a physical description.v[›] Another example are the Lorentz transformations, which relate measurements of time and velocity of two observers in motion relative to each other. They can be deduced in a purely group-theoretical way, by expressing the transformations as a rotational symmetry of Minkowski space. The latter serves—in the absence of significant gravitation—as a model of space time in special relativity. The full symmetry group of Minkowski space, i.e. including translations, is known as the Poincaré group. By the above, it plays a pivotal role in special relativity and, by implication, for quantum field theories. Symmetries that vary with location are central to the modern description of physical interactions with the help of gauge theory.


Question: What concept relates measurements of time and velocity of two observers in motion relative to each other?

Answer: Lorentz transformations

Lie groups are of fundamental importance in modern physics: Noether's theorem links continuous symmetries to conserved quantities. Rotation, as well as translations in space and time are basic symmetries of the laws of mechanics. <hl> They can, for instance, be used to construct simple models—imposing, say, axial symmetry on a situation will typically lead to significant simplification in the equations one needs to solve to provide a physical description.v[›] Another example are the Lorentz transformations , which relate measurements of time and velocity of two observers in motion relative to each other. <hl> They can be deduced in a purely group-theoretical way, by expressing the transformations as a rotational symmetry of Minkowski space. The latter serves—in the absence of significant gravitation—as a model of space time in special relativity. The full symmetry group of Minkowski space, i.e. including translations, is known as the Poincaré group. By the above, it plays a pivotal role in special relativity and, by implication, for quantum field theories. Symmetries that vary with location are central to the modern description of physical interactions with the help of gauge theory.

Lie groups are of fundamental importance in modern physics: Noether's theorem links continuous symmetries to conserved quantities. Rotation, as well as translations in space and time are basic symmetries of the laws of mechanics. They can, for instance, be used to construct simple models—imposing, say, axial symmetry on a situation will typically lead to significant simplification in the equations one needs to solve to provide a physical description.v[›] Another example are the <hl> Lorentz transformations <hl>, which relate measurements of time and velocity of two observers in motion relative to each other. They can be deduced in a purely group-theoretical way, by expressing the transformations as a rotational symmetry of Minkowski space. The latter serves—in the absence of significant gravitation—as a model of space time in special relativity. The full symmetry group of Minkowski space, i.e. including translations, is known as the Poincaré group. By the above, it plays a pivotal role in special relativity and, by implication, for quantum field theories. Symmetries that vary with location are central to the modern description of physical interactions with the help of gauge theory.

They can, for instance, be used to construct simple models—imposing, say, axial symmetry on a situation will typically lead to significant simplification in the equations one needs to solve to provide a physical description.v[›] Another example are the <hl> Lorentz transformations <hl> , which relate measurements of time and velocity of two observers in motion relative to each other.

question: What concept relates measurements of time and velocity of two observers in motion relative to each other?, context: Lie groups are of fundamental importance in modern physics: Noether's theorem links continuous symmetries to conserved quantities. Rotation, as well as translations in space and time are basic symmetries of the laws of mechanics. They can, for instance, be used to construct simple models—imposing, say, axial symmetry on a situation will typically lead to significant simplification in the equations one needs to solve to provide a physical description.v[›] Another example are the Lorentz transformations, which relate measurements of time and velocity of two observers in motion relative to each other. They can be deduced in a purely group-theoretical way, by expressing the transformations as a rotational symmetry of Minkowski space. The latter serves—in the absence of significant gravitation—as a model of space time in special relativity. The full symmetry group of Minkowski space, i.e. including translations, is known as the Poincaré group. By the above, it plays a pivotal role in special relativity and, by implication, for quantum field theories. Symmetries that vary with location are central to the modern description of physical interactions with the help of gauge theory.


Question: What describe the complete symmetry group of Minkowski space including translations?

Answer: Poincaré group

Lie groups are of fundamental importance in modern physics: Noether's theorem links continuous symmetries to conserved quantities. Rotation, as well as translations in space and time are basic symmetries of the laws of mechanics. They can, for instance, be used to construct simple models—imposing, say, axial symmetry on a situation will typically lead to significant simplification in the equations one needs to solve to provide a physical description.v[›] Another example are the Lorentz transformations, which relate measurements of time and velocity of two observers in motion relative to each other. They can be deduced in a purely group-theoretical way, by expressing the transformations as a rotational symmetry of Minkowski space. The latter serves—in the absence of significant gravitation—as a model of space time in special relativity. <hl> The full symmetry group of Minkowski space, i.e. including translations, is known as the Poincaré group . <hl> By the above, it plays a pivotal role in special relativity and, by implication, for quantum field theories. Symmetries that vary with location are central to the modern description of physical interactions with the help of gauge theory.

Lie groups are of fundamental importance in modern physics: Noether's theorem links continuous symmetries to conserved quantities. Rotation, as well as translations in space and time are basic symmetries of the laws of mechanics. They can, for instance, be used to construct simple models—imposing, say, axial symmetry on a situation will typically lead to significant simplification in the equations one needs to solve to provide a physical description.v[›] Another example are the Lorentz transformations, which relate measurements of time and velocity of two observers in motion relative to each other. They can be deduced in a purely group-theoretical way, by expressing the transformations as a rotational symmetry of Minkowski space. The latter serves—in the absence of significant gravitation—as a model of space time in special relativity. The full symmetry group of Minkowski space, i.e. including translations, is known as the <hl> Poincaré group <hl>. By the above, it plays a pivotal role in special relativity and, by implication, for quantum field theories. Symmetries that vary with location are central to the modern description of physical interactions with the help of gauge theory.

The full symmetry group of Minkowski space, i.e. including translations, is known as the <hl> Poincaré group <hl> .

question: What describe the complete symmetry group of Minkowski space including translations?, context: Lie groups are of fundamental importance in modern physics: Noether's theorem links continuous symmetries to conserved quantities. Rotation, as well as translations in space and time are basic symmetries of the laws of mechanics. They can, for instance, be used to construct simple models—imposing, say, axial symmetry on a situation will typically lead to significant simplification in the equations one needs to solve to provide a physical description.v[›] Another example are the Lorentz transformations, which relate measurements of time and velocity of two observers in motion relative to each other. They can be deduced in a purely group-theoretical way, by expressing the transformations as a rotational symmetry of Minkowski space. The latter serves—in the absence of significant gravitation—as a model of space time in special relativity. The full symmetry group of Minkowski space, i.e. including translations, is known as the Poincaré group. By the above, it plays a pivotal role in special relativity and, by implication, for quantum field theories. Symmetries that vary with location are central to the modern description of physical interactions with the help of gauge theory.


Question: What product is created if the requirement that every element has an inverse is eliminated?

Answer: monoid

In abstract algebra, more general structures are defined by relaxing some of the axioms defining a group. <hl> For example, if the requirement that every element has an inverse is eliminated, the resulting algebraic structure is called a monoid . <hl> The natural numbers N (including 0) under addition form a monoid, as do the nonzero integers under multiplication (Z ∖ {0}, ·), see above. There is a general method to formally add inverses to elements to any (abelian) monoid, much the same way as (Q ∖ {0}, ·) is derived from (Z ∖ {0}, ·), known as the Grothendieck group. Groupoids are similar to groups except that the composition a • b need not be defined for all a and b. They arise in the study of more complicated forms of symmetry, often in topological and analytical structures, such as the fundamental groupoid or stacks. Finally, it is possible to generalize any of these concepts by replacing the binary operation with an arbitrary n-ary one (i.e. an operation taking n arguments). With the proper generalization of the group axioms this gives rise to an n-ary group. The table gives a list of several structures generalizing groups.

In abstract algebra, more general structures are defined by relaxing some of the axioms defining a group. For example, if the requirement that every element has an inverse is eliminated, the resulting algebraic structure is called a <hl> monoid <hl>. The natural numbers N (including 0) under addition form a monoid, as do the nonzero integers under multiplication (Z ∖ {0}, ·), see above. There is a general method to formally add inverses to elements to any (abelian) monoid, much the same way as (Q ∖ {0}, ·) is derived from (Z ∖ {0}, ·), known as the Grothendieck group. Groupoids are similar to groups except that the composition a • b need not be defined for all a and b. They arise in the study of more complicated forms of symmetry, often in topological and analytical structures, such as the fundamental groupoid or stacks. Finally, it is possible to generalize any of these concepts by replacing the binary operation with an arbitrary n-ary one (i.e. an operation taking n arguments). With the proper generalization of the group axioms this gives rise to an n-ary group. The table gives a list of several structures generalizing groups.

For example, if the requirement that every element has an inverse is eliminated, the resulting algebraic structure is called a <hl> monoid <hl> .

question: What product is created if the requirement that every element has an inverse is eliminated?, context: In abstract algebra, more general structures are defined by relaxing some of the axioms defining a group. For example, if the requirement that every element has an inverse is eliminated, the resulting algebraic structure is called a monoid. The natural numbers N (including 0) under addition form a monoid, as do the nonzero integers under multiplication (Z ∖ {0}, ·), see above. There is a general method to formally add inverses to elements to any (abelian) monoid, much the same way as (Q ∖ {0}, ·) is derived from (Z ∖ {0}, ·), known as the Grothendieck group. Groupoids are similar to groups except that the composition a • b need not be defined for all a and b. They arise in the study of more complicated forms of symmetry, often in topological and analytical structures, such as the fundamental groupoid or stacks. Finally, it is possible to generalize any of these concepts by replacing the binary operation with an arbitrary n-ary one (i.e. an operation taking n arguments). With the proper generalization of the group axioms this gives rise to an n-ary group. The table gives a list of several structures generalizing groups.


Question: What type of numbers under addition form a monoid?

Answer: natural numbers N (including 0)

In abstract algebra, more general structures are defined by relaxing some of the axioms defining a group. For example, if the requirement that every element has an inverse is eliminated, the resulting algebraic structure is called a monoid. <hl> The natural numbers N (including 0) under addition form a monoid, as do the nonzero integers under multiplication (Z ∖ {0}, ·), see above. <hl> There is a general method to formally add inverses to elements to any (abelian) monoid, much the same way as (Q ∖ {0}, ·) is derived from (Z ∖ {0}, ·), known as the Grothendieck group. Groupoids are similar to groups except that the composition a • b need not be defined for all a and b. They arise in the study of more complicated forms of symmetry, often in topological and analytical structures, such as the fundamental groupoid or stacks. Finally, it is possible to generalize any of these concepts by replacing the binary operation with an arbitrary n-ary one (i.e. an operation taking n arguments). With the proper generalization of the group axioms this gives rise to an n-ary group. The table gives a list of several structures generalizing groups.

In abstract algebra, more general structures are defined by relaxing some of the axioms defining a group. For example, if the requirement that every element has an inverse is eliminated, the resulting algebraic structure is called a monoid. The <hl> natural numbers N (including 0) <hl> under addition form a monoid, as do the nonzero integers under multiplication (Z ∖ {0}, ·), see above. There is a general method to formally add inverses to elements to any (abelian) monoid, much the same way as (Q ∖ {0}, ·) is derived from (Z ∖ {0}, ·), known as the Grothendieck group. Groupoids are similar to groups except that the composition a • b need not be defined for all a and b. They arise in the study of more complicated forms of symmetry, often in topological and analytical structures, such as the fundamental groupoid or stacks. Finally, it is possible to generalize any of these concepts by replacing the binary operation with an arbitrary n-ary one (i.e. an operation taking n arguments). With the proper generalization of the group axioms this gives rise to an n-ary group. The table gives a list of several structures generalizing groups.

question: What type of numbers under addition form a monoid?, context: In abstract algebra, more general structures are defined by relaxing some of the axioms defining a group. For example, if the requirement that every element has an inverse is eliminated, the resulting algebraic structure is called a monoid. The natural numbers N (including 0) under addition form a monoid, as do the nonzero integers under multiplication (Z ∖ {0}, ·), see above. There is a general method to formally add inverses to elements to any (abelian) monoid, much the same way as (Q ∖ {0}, ·) is derived from (Z ∖ {0}, ·), known as the Grothendieck group. Groupoids are similar to groups except that the composition a • b need not be defined for all a and b. They arise in the study of more complicated forms of symmetry, often in topological and analytical structures, such as the fundamental groupoid or stacks. Finally, it is possible to generalize any of these concepts by replacing the binary operation with an arbitrary n-ary one (i.e. an operation taking n arguments). With the proper generalization of the group axioms this gives rise to an n-ary group. The table gives a list of several structures generalizing groups.


Question: What method formally adds inverses to elements to any monoid?

Answer: the Grothendieck group

In abstract algebra, more general structures are defined by relaxing some of the axioms defining a group. For example, if the requirement that every element has an inverse is eliminated, the resulting algebraic structure is called a monoid. The natural numbers N (including 0) under addition form a monoid, as do the nonzero integers under multiplication (Z ∖ {0}, ·), see above. <hl> There is a general method to formally add inverses to elements to any (abelian) monoid, much the same way as (Q ∖ {0}, ·) is derived from (Z ∖ {0}, ·), known as the Grothendieck group . <hl> Groupoids are similar to groups except that the composition a • b need not be defined for all a and b. They arise in the study of more complicated forms of symmetry, often in topological and analytical structures, such as the fundamental groupoid or stacks. Finally, it is possible to generalize any of these concepts by replacing the binary operation with an arbitrary n-ary one (i.e. an operation taking n arguments). With the proper generalization of the group axioms this gives rise to an n-ary group. The table gives a list of several structures generalizing groups.

In abstract algebra, more general structures are defined by relaxing some of the axioms defining a group. For example, if the requirement that every element has an inverse is eliminated, the resulting algebraic structure is called a monoid. The natural numbers N (including 0) under addition form a monoid, as do the nonzero integers under multiplication (Z ∖ {0}, ·), see above. There is a general method to formally add inverses to elements to any (abelian) monoid, much the same way as (Q ∖ {0}, ·) is derived from (Z ∖ {0}, ·), known as <hl> the Grothendieck group <hl>. Groupoids are similar to groups except that the composition a • b need not be defined for all a and b. They arise in the study of more complicated forms of symmetry, often in topological and analytical structures, such as the fundamental groupoid or stacks. Finally, it is possible to generalize any of these concepts by replacing the binary operation with an arbitrary n-ary one (i.e. an operation taking n arguments). With the proper generalization of the group axioms this gives rise to an n-ary group. The table gives a list of several structures generalizing groups.

There is a general method to formally add inverses to elements to any (abelian) monoid, much the same way as (Q ∖ {0}, ·) is derived from (Z ∖ {0}, ·), known as <hl> the Grothendieck group <hl> .

question: What method formally adds inverses to elements to any monoid?, context: In abstract algebra, more general structures are defined by relaxing some of the axioms defining a group. For example, if the requirement that every element has an inverse is eliminated, the resulting algebraic structure is called a monoid. The natural numbers N (including 0) under addition form a monoid, as do the nonzero integers under multiplication (Z ∖ {0}, ·), see above. There is a general method to formally add inverses to elements to any (abelian) monoid, much the same way as (Q ∖ {0}, ·) is derived from (Z ∖ {0}, ·), known as the Grothendieck group. Groupoids are similar to groups except that the composition a • b need not be defined for all a and b. They arise in the study of more complicated forms of symmetry, often in topological and analytical structures, such as the fundamental groupoid or stacks. Finally, it is possible to generalize any of these concepts by replacing the binary operation with an arbitrary n-ary one (i.e. an operation taking n arguments). With the proper generalization of the group axioms this gives rise to an n-ary group. The table gives a list of several structures generalizing groups.


Question: What can be replaced to simplify abstract algebra concepts?

Answer: the binary operation

In abstract algebra, more general structures are defined by relaxing some of the axioms defining a group. For example, if the requirement that every element has an inverse is eliminated, the resulting algebraic structure is called a monoid. The natural numbers N (including 0) under addition form a monoid, as do the nonzero integers under multiplication (Z ∖ {0}, ·), see above. There is a general method to formally add inverses to elements to any (abelian) monoid, much the same way as (Q ∖ {0}, ·) is derived from (Z ∖ {0}, ·), known as the Grothendieck group. Groupoids are similar to groups except that the composition a • b need not be defined for all a and b. They arise in the study of more complicated forms of symmetry, often in topological and analytical structures, such as the fundamental groupoid or stacks. <hl> Finally, it is possible to generalize any of these concepts by replacing the binary operation with an arbitrary n-ary one (i.e. an operation taking n arguments). <hl> With the proper generalization of the group axioms this gives rise to an n-ary group. The table gives a list of several structures generalizing groups.

In abstract algebra, more general structures are defined by relaxing some of the axioms defining a group. For example, if the requirement that every element has an inverse is eliminated, the resulting algebraic structure is called a monoid. The natural numbers N (including 0) under addition form a monoid, as do the nonzero integers under multiplication (Z ∖ {0}, ·), see above. There is a general method to formally add inverses to elements to any (abelian) monoid, much the same way as (Q ∖ {0}, ·) is derived from (Z ∖ {0}, ·), known as the Grothendieck group. Groupoids are similar to groups except that the composition a • b need not be defined for all a and b. They arise in the study of more complicated forms of symmetry, often in topological and analytical structures, such as the fundamental groupoid or stacks. Finally, it is possible to generalize any of these concepts by replacing <hl> the binary operation <hl> with an arbitrary n-ary one (i.e. an operation taking n arguments). With the proper generalization of the group axioms this gives rise to an n-ary group. The table gives a list of several structures generalizing groups.

question: What can be replaced to simplify abstract algebra concepts?, context: In abstract algebra, more general structures are defined by relaxing some of the axioms defining a group. For example, if the requirement that every element has an inverse is eliminated, the resulting algebraic structure is called a monoid. The natural numbers N (including 0) under addition form a monoid, as do the nonzero integers under multiplication (Z ∖ {0}, ·), see above. There is a general method to formally add inverses to elements to any (abelian) monoid, much the same way as (Q ∖ {0}, ·) is derived from (Z ∖ {0}, ·), known as the Grothendieck group. Groupoids are similar to groups except that the composition a • b need not be defined for all a and b. They arise in the study of more complicated forms of symmetry, often in topological and analytical structures, such as the fundamental groupoid or stacks. Finally, it is possible to generalize any of these concepts by replacing the binary operation with an arbitrary n-ary one (i.e. an operation taking n arguments). With the proper generalization of the group axioms this gives rise to an n-ary group. The table gives a list of several structures generalizing groups.


Question: Where is the Central African Republic located?

Answer: Central Africa

<hl> The Central Africa n Republic (CAR; Sango: Ködörösêse tî Bêafrîka; French: République centrafricaine pronounced: [ʁepyblik sɑ̃tʁafʁikɛn], or <hl> Centrafrique [sɑ̃tʀafʁik]) is a landlocked country in Central Africa. It is bordered by Chad to the north, Sudan to the northeast, South Sudan to the east, the Democratic Republic of the Congo and the Republic of the Congo to the south and Cameroon to the west. The CAR covers a land area of about 620,000 square kilometres (240,000 sq mi) and had an estimated population of around 4.7 million as of 2014[update].

The <hl> Central Africa <hl>n Republic (CAR; Sango: Ködörösêse tî Bêafrîka; French: République centrafricaine pronounced: [ʁepyblik sɑ̃tʁafʁikɛn], or Centrafrique [sɑ̃tʀafʁik]) is a landlocked country in Central Africa. It is bordered by Chad to the north, Sudan to the northeast, South Sudan to the east, the Democratic Republic of the Congo and the Republic of the Congo to the south and Cameroon to the west. The CAR covers a land area of about 620,000 square kilometres (240,000 sq mi) and had an estimated population of around 4.7 million as of 2014[update].

The <hl> Central Africa <hl> n Republic (CAR; Sango: Ködörösêse tî Bêafrîka; French: République centrafricaine pronounced: [ʁepyblik sɑ̃tʁafʁikɛn], or

question: Where is the Central African Republic located?, context: The Central African Republic (CAR; Sango: Ködörösêse tî Bêafrîka; French: République centrafricaine pronounced: [ʁepyblik sɑ̃tʁafʁikɛn], or Centrafrique [sɑ̃tʀafʁik]) is a landlocked country in Central Africa. It is bordered by Chad to the north, Sudan to the northeast, South Sudan to the east, the Democratic Republic of the Congo and the Republic of the Congo to the south and Cameroon to the west. The CAR covers a land area of about 620,000 square kilometres (240,000 sq mi) and had an estimated population of around 4.7 million as of 2014[update].


Question: What country borders CAR in the North?

Answer: Chad

The Central African Republic (CAR; Sango: Ködörösêse tî Bêafrîka; French: République centrafricaine pronounced: [ʁepyblik sɑ̃tʁafʁikɛn], or Centrafrique [sɑ̃tʀafʁik]) is a landlocked country in Central Africa. <hl> It is bordered by Chad to the north, Sudan to the northeast, South Sudan to the east, the Democratic Republic of the Congo and the Republic of the Congo to the south and Cameroon to the west. <hl> The CAR covers a land area of about 620,000 square kilometres (240,000 sq mi) and had an estimated population of around 4.7 million as of 2014[update].

The Central African Republic (CAR; Sango: Ködörösêse tî Bêafrîka; French: République centrafricaine pronounced: [ʁepyblik sɑ̃tʁafʁikɛn], or Centrafrique [sɑ̃tʀafʁik]) is a landlocked country in Central Africa. It is bordered by <hl> Chad <hl> to the north, Sudan to the northeast, South Sudan to the east, the Democratic Republic of the Congo and the Republic of the Congo to the south and Cameroon to the west. The CAR covers a land area of about 620,000 square kilometres (240,000 sq mi) and had an estimated population of around 4.7 million as of 2014[update].

question: What country borders CAR in the North?, context: The Central African Republic (CAR; Sango: Ködörösêse tî Bêafrîka; French: République centrafricaine pronounced: [ʁepyblik sɑ̃tʁafʁikɛn], or Centrafrique [sɑ̃tʀafʁik]) is a landlocked country in Central Africa. It is bordered by Chad to the north, Sudan to the northeast, South Sudan to the east, the Democratic Republic of the Congo and the Republic of the Congo to the south and Cameroon to the west. The CAR covers a land area of about 620,000 square kilometres (240,000 sq mi) and had an estimated population of around 4.7 million as of 2014[update].


Question: What Country borders CAR in the South?

Answer: the Republic of the Congo

The Central African Republic (CAR; Sango: Ködörösêse tî Bêafrîka; French: République centrafricaine pronounced: [ʁepyblik sɑ̃tʁafʁikɛn], or Centrafrique [sɑ̃tʀafʁik]) is a landlocked country in Central Africa. <hl> It is bordered by Chad to the north, Sudan to the northeast, South Sudan to the east, the Democratic Republic of the Congo and the Republic of the Congo to the south and Cameroon to the west. <hl> The CAR covers a land area of about 620,000 square kilometres (240,000 sq mi) and had an estimated population of around 4.7 million as of 2014[update].

The Central African Republic (CAR; Sango: Ködörösêse tî Bêafrîka; French: République centrafricaine pronounced: [ʁepyblik sɑ̃tʁafʁikɛn], or Centrafrique [sɑ̃tʀafʁik]) is a landlocked country in Central Africa. It is bordered by Chad to the north, Sudan to the northeast, South Sudan to the east, the Democratic Republic of the Congo and <hl> the Republic of the Congo <hl> to the south and Cameroon to the west. The CAR covers a land area of about 620,000 square kilometres (240,000 sq mi) and had an estimated population of around 4.7 million as of 2014[update].

question: What Country borders CAR in the South?, context: The Central African Republic (CAR; Sango: Ködörösêse tî Bêafrîka; French: République centrafricaine pronounced: [ʁepyblik sɑ̃tʁafʁikɛn], or Centrafrique [sɑ̃tʀafʁik]) is a landlocked country in Central Africa. It is bordered by Chad to the north, Sudan to the northeast, South Sudan to the east, the Democratic Republic of the Congo and the Republic of the Congo to the south and Cameroon to the west. The CAR covers a land area of about 620,000 square kilometres (240,000 sq mi) and had an estimated population of around 4.7 million as of 2014[update].


Question: What Country borders CAR in the West?

Answer: Cameroon

The Central African Republic (CAR; Sango: Ködörösêse tî Bêafrîka; French: République centrafricaine pronounced: [ʁepyblik sɑ̃tʁafʁikɛn], or Centrafrique [sɑ̃tʀafʁik]) is a landlocked country in Central Africa. <hl> It is bordered by Chad to the north, Sudan to the northeast, South Sudan to the east, the Democratic Republic of the Congo and the Republic of the Congo to the south and Cameroon to the west. <hl> The CAR covers a land area of about 620,000 square kilometres (240,000 sq mi) and had an estimated population of around 4.7 million as of 2014[update].

The Central African Republic (CAR; Sango: Ködörösêse tî Bêafrîka; French: République centrafricaine pronounced: [ʁepyblik sɑ̃tʁafʁikɛn], or Centrafrique [sɑ̃tʀafʁik]) is a landlocked country in Central Africa. It is bordered by Chad to the north, Sudan to the northeast, South Sudan to the east, the Democratic Republic of the Congo and the Republic of the Congo to the south and <hl> Cameroon <hl> to the west. The CAR covers a land area of about 620,000 square kilometres (240,000 sq mi) and had an estimated population of around 4.7 million as of 2014[update].

question: What Country borders CAR in the West?, context: The Central African Republic (CAR; Sango: Ködörösêse tî Bêafrîka; French: République centrafricaine pronounced: [ʁepyblik sɑ̃tʁafʁikɛn], or Centrafrique [sɑ̃tʀafʁik]) is a landlocked country in Central Africa. It is bordered by Chad to the north, Sudan to the northeast, South Sudan to the east, the Democratic Republic of the Congo and the Republic of the Congo to the south and Cameroon to the west. The CAR covers a land area of about 620,000 square kilometres (240,000 sq mi) and had an estimated population of around 4.7 million as of 2014[update].


Question: What is the population of CAR approximately?

Answer: 4.7 million

The Central African Republic (CAR; Sango: Ködörösêse tî Bêafrîka; French: République centrafricaine pronounced: [ʁepyblik sɑ̃tʁafʁikɛn], or Centrafrique [sɑ̃tʀafʁik]) is a landlocked country in Central Africa. It is bordered by Chad to the north, Sudan to the northeast, South Sudan to the east, the Democratic Republic of the Congo and the Republic of the Congo to the south and Cameroon to the west. <hl> The CAR covers a land area of about 620,000 square kilometres (240,000 sq mi) and had an estimated population of around 4.7 million as of 2014[update]. <hl>

The Central African Republic (CAR; Sango: Ködörösêse tî Bêafrîka; French: République centrafricaine pronounced: [ʁepyblik sɑ̃tʁafʁikɛn], or Centrafrique [sɑ̃tʀafʁik]) is a landlocked country in Central Africa. It is bordered by Chad to the north, Sudan to the northeast, South Sudan to the east, the Democratic Republic of the Congo and the Republic of the Congo to the south and Cameroon to the west. The CAR covers a land area of about 620,000 square kilometres (240,000 sq mi) and had an estimated population of around <hl> 4.7 million <hl> as of 2014[update].

question: What is the population of CAR approximately?, context: The Central African Republic (CAR; Sango: Ködörösêse tî Bêafrîka; French: République centrafricaine pronounced: [ʁepyblik sɑ̃tʁafʁikɛn], or Centrafrique [sɑ̃tʀafʁik]) is a landlocked country in Central Africa. It is bordered by Chad to the north, Sudan to the northeast, South Sudan to the east, the Democratic Republic of the Congo and the Republic of the Congo to the south and Cameroon to the west. The CAR covers a land area of about 620,000 square kilometres (240,000 sq mi) and had an estimated population of around 4.7 million as of 2014[update].


Question: Who established CAR's borders?

Answer: France

<hl> What is today the Central African Republic has been inhabited for millennia; however, the country's current borders were established by France , which ruled the country as a colony starting in the late 19th century. <hl> After gaining independence from France in 1960, the Central African Republic was ruled by a series of autocratic leaders; by the 1990s, calls for democracy led to the first multi-party democratic elections in 1993. Ange-Félix Patassé became president, but was later removed by General François Bozizé in the 2003 coup. The Central African Republic Bush War began in 2004 and, despite a peace treaty in 2007 and another in 2011, fighting broke out between various factions in December 2012, leading to ethnic and religious cleansing of the Muslim minority and massive population displacement in 2013 and 2014.

What is today the Central African Republic has been inhabited for millennia; however, the country's current borders were established by <hl> France <hl>, which ruled the country as a colony starting in the late 19th century. After gaining independence from France in 1960, the Central African Republic was ruled by a series of autocratic leaders; by the 1990s, calls for democracy led to the first multi-party democratic elections in 1993. Ange-Félix Patassé became president, but was later removed by General François Bozizé in the 2003 coup. The Central African Republic Bush War began in 2004 and, despite a peace treaty in 2007 and another in 2011, fighting broke out between various factions in December 2012, leading to ethnic and religious cleansing of the Muslim minority and massive population displacement in 2013 and 2014.

What is today the Central African Republic has been inhabited for millennia; however, the country's current borders were established by <hl> France <hl> , which ruled the country as a colony starting in the late 19th century.

question: Who established CAR's borders?, context: What is today the Central African Republic has been inhabited for millennia; however, the country's current borders were established by France, which ruled the country as a colony starting in the late 19th century. After gaining independence from France in 1960, the Central African Republic was ruled by a series of autocratic leaders; by the 1990s, calls for democracy led to the first multi-party democratic elections in 1993. Ange-Félix Patassé became president, but was later removed by General François Bozizé in the 2003 coup. The Central African Republic Bush War began in 2004 and, despite a peace treaty in 2007 and another in 2011, fighting broke out between various factions in December 2012, leading to ethnic and religious cleansing of the Muslim minority and massive population displacement in 2013 and 2014.


Question: What country ruled CAR as a colony?

Answer: France

<hl> What is today the Central African Republic has been inhabited for millennia; however, the country's current borders were established by France , which ruled the country as a colony starting in the late 19th century. <hl> After gaining independence from France in 1960, the Central African Republic was ruled by a series of autocratic leaders; by the 1990s, calls for democracy led to the first multi-party democratic elections in 1993. Ange-Félix Patassé became president, but was later removed by General François Bozizé in the 2003 coup. The Central African Republic Bush War began in 2004 and, despite a peace treaty in 2007 and another in 2011, fighting broke out between various factions in December 2012, leading to ethnic and religious cleansing of the Muslim minority and massive population displacement in 2013 and 2014.

What is today the Central African Republic has been inhabited for millennia; however, the country's current borders were established by <hl> France <hl>, which ruled the country as a colony starting in the late 19th century. After gaining independence from France in 1960, the Central African Republic was ruled by a series of autocratic leaders; by the 1990s, calls for democracy led to the first multi-party democratic elections in 1993. Ange-Félix Patassé became president, but was later removed by General François Bozizé in the 2003 coup. The Central African Republic Bush War began in 2004 and, despite a peace treaty in 2007 and another in 2011, fighting broke out between various factions in December 2012, leading to ethnic and religious cleansing of the Muslim minority and massive population displacement in 2013 and 2014.

What is today the Central African Republic has been inhabited for millennia; however, the country's current borders were established by <hl> France <hl> , which ruled the country as a colony starting in the late 19th century.

question: What country ruled CAR as a colony?, context: What is today the Central African Republic has been inhabited for millennia; however, the country's current borders were established by France, which ruled the country as a colony starting in the late 19th century. After gaining independence from France in 1960, the Central African Republic was ruled by a series of autocratic leaders; by the 1990s, calls for democracy led to the first multi-party democratic elections in 1993. Ange-Félix Patassé became president, but was later removed by General François Bozizé in the 2003 coup. The Central African Republic Bush War began in 2004 and, despite a peace treaty in 2007 and another in 2011, fighting broke out between various factions in December 2012, leading to ethnic and religious cleansing of the Muslim minority and massive population displacement in 2013 and 2014.


Question: When did CAR become independent?

Answer: in 1960

What is today the Central African Republic has been inhabited for millennia; however, the country's current borders were established by France, which ruled the country as a colony starting in the late 19th century. <hl> After gaining independence from France in 1960 , the Central African Republic was ruled by a series of autocratic leaders; by the 1990s, calls for democracy led to the first multi-party democratic elections in 1993. <hl> Ange-Félix Patassé became president, but was later removed by General François Bozizé in the 2003 coup. The Central African Republic Bush War began in 2004 and, despite a peace treaty in 2007 and another in 2011, fighting broke out between various factions in December 2012, leading to ethnic and religious cleansing of the Muslim minority and massive population displacement in 2013 and 2014.

What is today the Central African Republic has been inhabited for millennia; however, the country's current borders were established by France, which ruled the country as a colony starting in the late 19th century. After gaining independence from France <hl> in 1960 <hl>, the Central African Republic was ruled by a series of autocratic leaders; by the 1990s, calls for democracy led to the first multi-party democratic elections in 1993. Ange-Félix Patassé became president, but was later removed by General François Bozizé in the 2003 coup. The Central African Republic Bush War began in 2004 and, despite a peace treaty in 2007 and another in 2011, fighting broke out between various factions in December 2012, leading to ethnic and religious cleansing of the Muslim minority and massive population displacement in 2013 and 2014.

After gaining independence from France <hl> in 1960 <hl> , the Central African Republic was ruled by a series of autocratic leaders; by the 1990s, calls for democracy led to the first multi-party democratic elections in 1993.

question: When did CAR become independent?, context: What is today the Central African Republic has been inhabited for millennia; however, the country's current borders were established by France, which ruled the country as a colony starting in the late 19th century. After gaining independence from France in 1960, the Central African Republic was ruled by a series of autocratic leaders; by the 1990s, calls for democracy led to the first multi-party democratic elections in 1993. Ange-Félix Patassé became president, but was later removed by General François Bozizé in the 2003 coup. The Central African Republic Bush War began in 2004 and, despite a peace treaty in 2007 and another in 2011, fighting broke out between various factions in December 2012, leading to ethnic and religious cleansing of the Muslim minority and massive population displacement in 2013 and 2014.


Question: Who was the first democratically elected President?

Answer: Ange-Félix Patassé

What is today the Central African Republic has been inhabited for millennia; however, the country's current borders were established by France, which ruled the country as a colony starting in the late 19th century. After gaining independence from France in 1960, the Central African Republic was ruled by a series of autocratic leaders; by the 1990s, calls for democracy led to the first multi-party democratic elections in 1993. <hl> Ange-Félix Patassé became president, but was later removed by General François Bozizé in the 2003 coup. <hl> The Central African Republic Bush War began in 2004 and, despite a peace treaty in 2007 and another in 2011, fighting broke out between various factions in December 2012, leading to ethnic and religious cleansing of the Muslim minority and massive population displacement in 2013 and 2014.

What is today the Central African Republic has been inhabited for millennia; however, the country's current borders were established by France, which ruled the country as a colony starting in the late 19th century. After gaining independence from France in 1960, the Central African Republic was ruled by a series of autocratic leaders; by the 1990s, calls for democracy led to the first multi-party democratic elections in 1993. <hl> Ange-Félix Patassé <hl> became president, but was later removed by General François Bozizé in the 2003 coup. The Central African Republic Bush War began in 2004 and, despite a peace treaty in 2007 and another in 2011, fighting broke out between various factions in December 2012, leading to ethnic and religious cleansing of the Muslim minority and massive population displacement in 2013 and 2014.

question: Who was the first democratically elected President?, context: What is today the Central African Republic has been inhabited for millennia; however, the country's current borders were established by France, which ruled the country as a colony starting in the late 19th century. After gaining independence from France in 1960, the Central African Republic was ruled by a series of autocratic leaders; by the 1990s, calls for democracy led to the first multi-party democratic elections in 1993. Ange-Félix Patassé became president, but was later removed by General François Bozizé in the 2003 coup. The Central African Republic Bush War began in 2004 and, despite a peace treaty in 2007 and another in 2011, fighting broke out between various factions in December 2012, leading to ethnic and religious cleansing of the Muslim minority and massive population displacement in 2013 and 2014.


Question: What religion is a minority in CAR?

Answer: Muslim minority

What is today the Central African Republic has been inhabited for millennia; however, the country's current borders were established by France, which ruled the country as a colony starting in the late 19th century. After gaining independence from France in 1960, the Central African Republic was ruled by a series of autocratic leaders; by the 1990s, calls for democracy led to the first multi-party democratic elections in 1993. Ange-Félix Patassé became president, but was later removed by General François Bozizé in the 2003 coup. <hl> The Central African Republic Bush War began in 2004 and, despite a peace treaty in 2007 and another in 2011, fighting broke out between various factions in December 2012, leading to ethnic and religious cleansing of the Muslim minority and massive population displacement in 2013 and 2014. <hl>

What is today the Central African Republic has been inhabited for millennia; however, the country's current borders were established by France, which ruled the country as a colony starting in the late 19th century. After gaining independence from France in 1960, the Central African Republic was ruled by a series of autocratic leaders; by the 1990s, calls for democracy led to the first multi-party democratic elections in 1993. Ange-Félix Patassé became president, but was later removed by General François Bozizé in the 2003 coup. The Central African Republic Bush War began in 2004 and, despite a peace treaty in 2007 and another in 2011, fighting broke out between various factions in December 2012, leading to ethnic and religious cleansing of the <hl> Muslim minority <hl> and massive population displacement in 2013 and 2014.

question: What religion is a minority in CAR?, context: What is today the Central African Republic has been inhabited for millennia; however, the country's current borders were established by France, which ruled the country as a colony starting in the late 19th century. After gaining independence from France in 1960, the Central African Republic was ruled by a series of autocratic leaders; by the 1990s, calls for democracy led to the first multi-party democratic elections in 1993. Ange-Félix Patassé became president, but was later removed by General François Bozizé in the 2003 coup. The Central African Republic Bush War began in 2004 and, despite a peace treaty in 2007 and another in 2011, fighting broke out between various factions in December 2012, leading to ethnic and religious cleansing of the Muslim minority and massive population displacement in 2013 and 2014.


Question: How long ago did people start living in the area?

Answer: Approximately 10,000 years ago

<hl> Approximately 10,000 years ago , desertification forced hunter-gatherer societies south into the Sahel regions of northern Central Africa, where some groups settled and began farming as part of the Neolithic Revolution. <hl> Initial farming of white yam progressed into millet and sorghum, and before 3000 BC the domestication of African oil palm improved the groups' nutrition and allowed for expansion of the local populations. Bananas arrived in the region and added an important source of carbohydrates to the diet; they were also used in the production of alcoholic beverages.[when?] This Agricultural Revolution, combined with a "Fish-stew Revolution", in which fishing began to take place, and the use of boats, allowed for the transportation of goods. Products were often moved in ceramic pots, which are the first known examples of artistic expression from the region's inhabitants.

<hl> Approximately 10,000 years ago <hl>, desertification forced hunter-gatherer societies south into the Sahel regions of northern Central Africa, where some groups settled and began farming as part of the Neolithic Revolution. Initial farming of white yam progressed into millet and sorghum, and before 3000 BC the domestication of African oil palm improved the groups' nutrition and allowed for expansion of the local populations. Bananas arrived in the region and added an important source of carbohydrates to the diet; they were also used in the production of alcoholic beverages.[when?] This Agricultural Revolution, combined with a "Fish-stew Revolution", in which fishing began to take place, and the use of boats, allowed for the transportation of goods. Products were often moved in ceramic pots, which are the first known examples of artistic expression from the region's inhabitants.

<hl> Approximately 10,000 years ago <hl> , desertification forced hunter-gatherer societies south into the Sahel regions of northern Central Africa, where some groups settled and began farming as part of the Neolithic Revolution.

question: How long ago did people start living in the area?, context: Approximately 10,000 years ago, desertification forced hunter-gatherer societies south into the Sahel regions of northern Central Africa, where some groups settled and began farming as part of the Neolithic Revolution. Initial farming of white yam progressed into millet and sorghum, and before 3000 BC the domestication of African oil palm improved the groups' nutrition and allowed for expansion of the local populations. Bananas arrived in the region and added an important source of carbohydrates to the diet; they were also used in the production of alcoholic beverages.[when?] This Agricultural Revolution, combined with a "Fish-stew Revolution", in which fishing began to take place, and the use of boats, allowed for the transportation of goods. Products were often moved in ceramic pots, which are the first known examples of artistic expression from the region's inhabitants.


Question: What agricultural job did people settling in the area perform?

Answer: farming

<hl> Approximately 10,000 years ago, desertification forced hunter-gatherer societies south into the Sahel regions of northern Central Africa, where some groups settled and began farming as part of the Neolithic Revolution. <hl> Initial farming of white yam progressed into millet and sorghum, and before 3000 BC the domestication of African oil palm improved the groups' nutrition and allowed for expansion of the local populations. Bananas arrived in the region and added an important source of carbohydrates to the diet; they were also used in the production of alcoholic beverages.[when?] This Agricultural Revolution, combined with a "Fish-stew Revolution", in which fishing began to take place, and the use of boats, allowed for the transportation of goods. Products were often moved in ceramic pots, which are the first known examples of artistic expression from the region's inhabitants.

Approximately 10,000 years ago, desertification forced hunter-gatherer societies south into the Sahel regions of northern Central Africa, where some groups settled and began <hl> farming <hl> as part of the Neolithic Revolution. Initial farming of white yam progressed into millet and sorghum, and before 3000 BC the domestication of African oil palm improved the groups' nutrition and allowed for expansion of the local populations. Bananas arrived in the region and added an important source of carbohydrates to the diet; they were also used in the production of alcoholic beverages.[when?] This Agricultural Revolution, combined with a "Fish-stew Revolution", in which fishing began to take place, and the use of boats, allowed for the transportation of goods. Products were often moved in ceramic pots, which are the first known examples of artistic expression from the region's inhabitants.

question: What agricultural job did people settling in the area perform?, context: Approximately 10,000 years ago, desertification forced hunter-gatherer societies south into the Sahel regions of northern Central Africa, where some groups settled and began farming as part of the Neolithic Revolution. Initial farming of white yam progressed into millet and sorghum, and before 3000 BC the domestication of African oil palm improved the groups' nutrition and allowed for expansion of the local populations. Bananas arrived in the region and added an important source of carbohydrates to the diet; they were also used in the production of alcoholic beverages.[when?] This Agricultural Revolution, combined with a "Fish-stew Revolution", in which fishing began to take place, and the use of boats, allowed for the transportation of goods. Products were often moved in ceramic pots, which are the first known examples of artistic expression from the region's inhabitants.


Question: What vegetation helped populations expand?

Answer: African oil palm

Approximately 10,000 years ago, desertification forced hunter-gatherer societies south into the Sahel regions of northern Central Africa, where some groups settled and began farming as part of the Neolithic Revolution. <hl> Initial farming of white yam progressed into millet and sorghum, and before 3000 BC the domestication of African oil palm improved the groups' nutrition and allowed for expansion of the local populations. <hl> Bananas arrived in the region and added an important source of carbohydrates to the diet; they were also used in the production of alcoholic beverages.[when?] This Agricultural Revolution, combined with a "Fish-stew Revolution", in which fishing began to take place, and the use of boats, allowed for the transportation of goods. Products were often moved in ceramic pots, which are the first known examples of artistic expression from the region's inhabitants.

Approximately 10,000 years ago, desertification forced hunter-gatherer societies south into the Sahel regions of northern Central Africa, where some groups settled and began farming as part of the Neolithic Revolution. Initial farming of white yam progressed into millet and sorghum, and before 3000 BC the domestication of <hl> African oil palm <hl> improved the groups' nutrition and allowed for expansion of the local populations. Bananas arrived in the region and added an important source of carbohydrates to the diet; they were also used in the production of alcoholic beverages.[when?] This Agricultural Revolution, combined with a "Fish-stew Revolution", in which fishing began to take place, and the use of boats, allowed for the transportation of goods. Products were often moved in ceramic pots, which are the first known examples of artistic expression from the region's inhabitants.

question: What vegetation helped populations expand?, context: Approximately 10,000 years ago, desertification forced hunter-gatherer societies south into the Sahel regions of northern Central Africa, where some groups settled and began farming as part of the Neolithic Revolution. Initial farming of white yam progressed into millet and sorghum, and before 3000 BC the domestication of African oil palm improved the groups' nutrition and allowed for expansion of the local populations. Bananas arrived in the region and added an important source of carbohydrates to the diet; they were also used in the production of alcoholic beverages.[when?] This Agricultural Revolution, combined with a "Fish-stew Revolution", in which fishing began to take place, and the use of boats, allowed for the transportation of goods. Products were often moved in ceramic pots, which are the first known examples of artistic expression from the region's inhabitants.