nc33/MultiSpan_SQUAD huggingface dataset:
Group_(mathematics)

The University of Chicago's 1960–61 Group Theory Year brought together group theorists such as Daniel Gorenstein, John G. Thompson and Walter Feit, laying the foundation of a collaboration that, with input from numerous other mathematicians, classified all finite simple groups in 1982. This project exceeded previous mathematical endeavours by its sheer size, in both length of proof and number of researchers. Research is ongoing to simplify the proof of this classification. These days, group theory is still a highly active mathematical branch, impacting many other fields.a[›]

Question: Where did the group of theorists first meet?
Answer: The University of Chicago

Question: What university class year belonged to the group of theorists?
Answer: 1960–61

Question: What did the group of theorists classify in 1982?
Answer: all finite simple groups

Question: What two factors describe the large size of the project?
Answer: length of proof and number of researchers.
Group_(mathematics)

To understand groups beyond the level of mere symbolic manipulations as above, more structural concepts have to be employed.c[›] There is a conceptual principle underlying all of the following notions: to take advantage of the structure offered by groups (which sets, being "structureless", do not have), constructions related to groups have to be compatible with the group operation. This compatibility manifests itself in the following notions in various ways. For example, groups can be related to each other via functions called group homomorphisms. By the mentioned principle, they are required to respect the group structures in a precise sense. The structure of groups can also be understood by breaking them into pieces called subgroups and quotient groups. The principle of "preserving structures"—a recurring topic in mathematics throughout—is an instance of working in a category, in this case the category of groups.

Question: What ideas are used to understand groups beyond symbols?
Answer: structural concepts

Question: What has to be compatible with the group operation?
Answer: constructions related to groups

Question: What concept describes groups that can be related to each other via functions?
Answer: group homomorphisms
Group_(mathematics)

Two groups G and H are called isomorphic if there exist group homomorphisms a: G → H and b: H → G, such that applying the two functions one after another in each of the two possible orders gives the identity functions of G and H. That is, a(b(h)) = h and b(a(g)) = g for any g in G and h in H. From an abstract point of view, isomorphic groups carry the same information. For example, proving that g • g = 1G for some element g of G is equivalent to proving that a(g) ∗ a(g) = 1H, because applying a to the first equality yields the second, and applying b to the second gives back the first.

Question: What are two groups called if they include homomorphisms?
Answer: isomorphic

Question: What similar element do isomorphic groups carry?
Answer: isomorphic groups

Question: How can showing the second equality yields the first prove the concept of isomorphic groups?
Answer: applying a to the first equality yields the second
Group_(mathematics)

In the example above, the identity and the rotations constitute a subgroup R = {id, r1, r2, r3}, highlighted in red in the group table above: any two rotations composed are still a rotation, and a rotation can be undone by (i.e. is inverse to) the complementary rotations 270° for 90°, 180° for 180°, and 90° for 270° (note that rotation in the opposite direction is not defined). The subgroup test is a necessary and sufficient condition for a subset H of a group G to be a subgroup: it is sufficient to check that g−1h ∈ H for all elements g, h ∈ H. Knowing the subgroups is important in understanding the group as a whole.d[›]

Question: What is composed of two rotations?
Answer: a rotation

Question: What rotation can a rotation be reversed by?
Answer: inverse

Question: What essential condition must be met for a subset of a group to be a subgroup?
Answer: The subgroup test
Group_(mathematics)

In many situations it is desirable to consider two group elements the same if they differ by an element of a given subgroup. For example, in D4 above, once a reflection is performed, the square never gets back to the r2 configuration by just applying the rotation operations (and no further reflections), i.e. the rotation operations are irrelevant to the question whether a reflection has been performed. Cosets are used to formalize this insight: a subgroup H defines left and right cosets, which can be thought of as translations of H by arbitrary group elements g. In symbolic terms, the left and right cosets of H containing g are

Question: When is it preferable to consider two group elements to be the same?
Answer: irrelevant to the question whether a reflection has been performed

Question: What should not be considered when asking if a reflection has been performed?
Answer: rotation operations

Question: What number sets are used to show how subgroups can be seen as translations of the larger group?
Answer: Cosets
Group_(mathematics)

This set inherits a group operation (sometimes called coset multiplication, or coset addition) from the original group G: (gN) • (hN) = (gh)N for all g and h in G. This definition is motivated by the idea (itself an instance of general structural considerations outlined above) that the map G → G / N that associates to any element g its coset gN be a group homomorphism, or by general abstract considerations called universal properties. The coset eN = N serves as the identity in this group, and the inverse of gN in the quotient group is (gN)−1 = (g−1)N.e[›]

Question: What is another term for coset multiplication?
Answer: coset addition

Question: What group operation is taken by the set from the original group?
Answer: coset multiplication

Question: What group does the inverse of gN include?
Answer: quotient group
Group_(mathematics)

Quotient groups and subgroups together form a way of describing every group by its presentation: any group is the quotient of the free group over the generators of the group, quotiented by the subgroup of relations. The dihedral group D4, for example, can be generated by two elements r and f (for example, r = r1, the right rotation and f = fv the vertical (or any other) reflection), which means that every symmetry of the square is a finite composition of these two symmetries or their inverses. Together with the relations

Question: What groups can be combined to describe every group?
Answer: is the quotient of the free group over the generators of the group

Question: What group can be the quotient of the free group over the generators of the group?
Answer: any group

Question: What is a finite composition of two symmetries or their inverses?
Answer: every symmetry of the square
Group_(mathematics)

Sub- and quotient groups are related in the following way: a subset H of G can be seen as an injective map H → G, i.e. any element of the target has at most one element that maps to it. The counterpart to injective maps are surjective maps (every element of the target is mapped onto), such as the canonical map G → G / N.y[›] Interpreting subgroup and quotients in light of these homomorphisms emphasizes the structural concept inherent to these definitions alluded to in the introduction. In general, homomorphisms are neither injective nor surjective. Kernel and image of group homomorphisms and the first isomorphism theorem address this phenomenon.

Question: What map shows the relation between sub and quotient groups?
Answer: injective map

Question: What are the opposites of injective maps?
Answer: surjective maps

Question: What is an example of a surjective map?
Answer: canonical map

Question: What theory address the phenomenon of homomorphisms being neither injective nor surjective?
Answer: the first isomorphism theorem
Group_(mathematics)

Groups are also applied in many other mathematical areas. Mathematical objects are often examined by associating groups to them and studying the properties of the corresponding groups. For example, Henri Poincaré founded what is now called algebraic topology by introducing the fundamental group. By means of this connection, topological properties such as proximity and continuity translate into properties of groups.i[›] For example, elements of the fundamental group are represented by loops. The second image at the right shows some loops in a plane minus a point. The blue loop is considered null-homotopic (and thus irrelevant), because it can be continuously shrunk to a point. The presence of the hole prevents the orange loop from being shrunk to a point. The fundamental group of the plane with a point deleted turns out to be infinite cyclic, generated by the orange loop (or any other loop winding once around the hole). This way, the fundamental group detects the hole.

Question: What are usually analyzed by associating groups to them and studying the elements of the corresponding groups?
Answer: Mathematical objects

Question: Who founded algebraic topology?
Answer: Henri Poincaré

Question: What did Henri Poincaré introduce when he established algebraic topology?
Answer: the fundamental group

Question: What topological properties translate into properties of groups?
Answer: proximity and continuity

Question: What represents elements of the fundamental group?
Answer: loops
Group_(mathematics)

In modular arithmetic, two integers are added and then the sum is divided by a positive integer called the modulus. The result of modular addition is the remainder of that division. For any modulus, n, the set of integers from 0 to n − 1 forms a group under modular addition: the inverse of any element a is n − a, and 0 is the identity element. This is familiar from the addition of hours on the face of a clock: if the hour hand is on 9 and is advanced 4 hours, it ends up on 1, as shown at the right. This is expressed by saying that 9 + 4 equals 1 "modulo 12" or, in symbols,

Question: What positive integer is used to divide the sum of two positive integers in modular mathematics?
Answer: the modulus

Question: What results from modular addition?
Answer: the remainder of that division

Question: What type of device can be use to demonstrate modular addition?
Answer: a clock
Group_(mathematics)

For any prime number p, there is also the multiplicative group of integers modulo p. Its elements are the integers 1 to p − 1. The group operation is multiplication modulo p. That is, the usual product is divided by p and the remainder of this division is the result of modular multiplication. For example, if p = 5, there are four group elements 1, 2, 3, 4. In this group, 4 · 4 = 1, because the usual product 16 is equivalent to 1, which divided by 5 yields a remainder of 1. for 5 divides 16 − 1 = 15, denoted

Question: What term describes the group of integers related to a prime number?
Answer: modulo p

Question: What integers are included in modulo p?
Answer: 1 to p − 1

Question: How many group elements exist if p=5?
Answer: four group elements
Group_(mathematics)

In the groups Z/nZ introduced above, the element 1 is primitive, so these groups are cyclic. Indeed, each element is expressible as a sum all of whose terms are 1. Any cyclic group with n elements is isomorphic to this group. A second example for cyclic groups is the group of n-th complex roots of unity, given by complex numbers z satisfying zn = 1. These numbers can be visualized as the vertices on a regular n-gon, as shown in blue at the right for n = 6. The group operation is multiplication of complex numbers. In the picture, multiplying with z corresponds to a counter-clockwise rotation by 60°. Using some field theory, the group Fp× can be shown to be cyclic: for example, if p = 5, 3 is a generator since 31 = 3, 32 = 9 ≡ 4, 33 ≡ 2, and 34 ≡ 1.

Question: What is a group labeled when the element 1 is primitive?
Answer: cyclic

Question: What group is isomorphic to cyclic groups?
Answer: Any cyclic group with n elements

Question: What example of cyclic group satisfies the express of zn = 1?
Answer: the group of n-th complex roots of unity
Group_(mathematics)

Symmetry groups are groups consisting of symmetries of given mathematical objects—be they of geometric nature, such as the introductory symmetry group of the square, or of algebraic nature, such as polynomial equations and their solutions. Conceptually, group theory can be thought of as the study of symmetry.t[›] Symmetries in mathematics greatly simplify the study of geometrical or analytical objects. A group is said to act on another mathematical object X if every group element performs some operation on X compatibly to the group law. In the rightmost example below, an element of order 7 of the (2,3,7) triangle group acts on the tiling by permuting the highlighted warped triangles (and the other ones, too). By a group action, the group pattern is connected to the structure of the object being acted on.

Question: What are groups consisting of symmetries of given arithmetic concepts?
Answer: Symmetry groups

Question: What term describes the introductory symmetry group of the square?
Answer: geometric nature

Question: What can regarded as the study of symmetry?
Answer: group theory

Question: What is the name of the rule that must be met for a group operation to occur?
Answer: group law

Question: The group pattern is connected to the structure of the target by what behavior?
Answer: group action
Group_(mathematics)

Likewise, group theory helps predict the changes in physical properties that occur when a material undergoes a phase transition, for example, from a cubic to a tetrahedral crystalline form. An example is ferroelectric materials, where the change from a paraelectric to a ferroelectric state occurs at the Curie temperature and is related to a change from the high-symmetry paraelectric state to the lower symmetry ferroelectic state, accompanied by a so-called soft phonon mode, a vibrational lattice mode that goes to zero frequency at the transition.

Question: What aids in predicting changes of physical traits?
Answer: group theory

Question: What stage of a physical transformation can group theory be used to make prediction?
Answer: phase transition

Question: What temperature causes the change of ferroelectric materials?
Answer: Curie temperature

Question: What term describes the vibrational lattice mode that turns to 0 frequency at the change?
Answer: soft phonon mode
Group_(mathematics)

Finite symmetry groups such as the Mathieu groups are used in coding theory, which is in turn applied in error correction of transmitted data, and in CD players. Another application is differential Galois theory, which characterizes functions having antiderivatives of a prescribed form, giving group-theoretic criteria for when solutions of certain differential equations are well-behaved.u[›] Geometric properties that remain stable under group actions are investigated in (geometric) invariant theory.

Question: What are examples of finite symmetry groups used in coding theory?
Answer: Mathieu groups

Question: What is used for error correction of transferred data?
Answer: coding theory

Question: What describes functions having antiderivatives of a prescribed form?
Answer: differential Galois theory

Question: What concept investigates geometric elements that stay stable under group action?
Answer: invariant theory
Group_(mathematics)

Matrix groups consist of matrices together with matrix multiplication. The general linear group GL(n, R) consists of all invertible n-by-n matrices with real entries. Its subgroups are referred to as matrix groups or linear groups. The dihedral group example mentioned above can be viewed as a (very small) matrix group. Another important matrix group is the special orthogonal group SO(n). It describes all possible rotations in n dimensions. Via Euler angles, rotation matrices are used in computer graphics.

Question: What groups combine matrices with matrix multiplication?
Answer: Matrix groups

Question: What includes all invertible n-by-n matrices with real entries?
Answer: The general linear group

Question: What term describes subgroups of the general linear group?
Answer: matrix groups

Question: What matrix group portrays all possible rotations in n dimensions?
Answer: the special orthogonal group SO(n)

Question: Rotation matrix groups are utilized in computer graphics with what concept?
Answer: Euler angles
Group_(mathematics)

Exchanging "+" and "−" in the expression, i.e. permuting the two solutions of the equation can be viewed as a (very simple) group operation. Similar formulae are known for cubic and quartic equations, but do not exist in general for degree 5 and higher. Abstract properties of Galois groups associated with polynomials (in particular their solvability) give a criterion for polynomials that have all their solutions expressible by radicals, i.e. solutions expressible using solely addition, multiplication, and roots similar to the formula above.

Question: What degree does not include simple formulas for cubic and quatric equations?
Answer: degree 5 and higher

Question: What concept is associated with the solvability of polynomials?
Answer: Abstract properties of Galois groups

Question: What are used to express the solutions of polynomials?
Answer: radicals
Group_(mathematics)

A group is called finite if it has a finite number of elements. The number of elements is called the order of the group. An important class is the symmetric groups SN, the groups of permutations of N letters. For example, the symmetric group on 3 letters S3 is the group consisting of all possible orderings of the three letters ABC, i.e. contains the elements ABC, ACB, ..., up to CBA, in total 6 (or 3 factorial) elements. This class is fundamental insofar as any finite group can be expressed as a subgroup of a symmetric group SN for a suitable integer N (Cayley's theorem). Parallel to the group of symmetries of the square above, S3 can also be interpreted as the group of symmetries of an equilateral triangle.

Question: What does a finite group include?
Answer: a finite number of elements

Question: What is the number of elements in a group named?
Answer: the order of the group

Question: What type of class has a finite group that can be expressed as a subgroup of a symmetric group?
Answer: fundamental

Question: What can be described as the group of symmetries of an equilateral triangle?
Answer: S3
Group_(mathematics)

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 level of finality do mathematicians try to reach with math concepts? 
Answer: complete classification

Question: What area of classification leads to troublesome arithmetic?
Answer: finite groups

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

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

Question: What describes finite simple groups as the building pieces for all finite groups?
Answer: The Jordan–Hölder theorem
Group_(mathematics)

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 group operations must occur for group law and topology to integrate well?
Answer: continuous functions

Question: What is the most simple example of topological groups?
Answer: reals R under addition, (R ∖ {0}, ·),

Question: What variables do locally compact groups share that can be studied by harmonic analysis?
Answer: Haar measures
Group_(mathematics)

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 concepts are fundamental to number theory?
Answer: adele rings and adelic algebraic groups

Question: What group uses infinite field extensions with topology?
Answer: the absolute Galois group

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

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