Group_(mathematics)

context: In mathematics, a group is an algebraic structure consisting of a set of elements equipped with an operation that combines any two elements to form a third element. The operation satisfies four conditions called the group axioms, namely closure, associativity, identity and invertibility. One of the most familiar examples of a group is the set of integers together with the addition operation, but the abstract formalization of the group axioms, detached as it is from the concrete nature of any particular group and its operation, applies much more widely. It allows entities with highly diverse mathematical origins in abstract algebra and beyond to be handled in a flexible way while retaining their essential structural aspects. The ubiquity of groups in numerous areas within and outside mathematics makes them a central organizing principle of contemporary mathematics.
qas: answers: a group
question: What is an arithmetical structure comprising of a set of elements including an operation that joins any two elements to form a third element?

answers: group axioms
question: What is the name used to describe the four conditions of closure, associativity, identity and invertibility?

answers: abstract formalization of the group axioms
question: What permits elements with different mathematical starting points to be taken care of in a flexible manner while holding their key basic aspects?

context: Groups share a fundamental kinship with the notion of symmetry. For example, a symmetry group encodes symmetry features of a geometrical object: the group consists of the set of transformations that leave the object unchanged and the operation of combining two such transformations by performing one after the other. Lie groups are the symmetry groups used in the Standard Model of particle physics; Point groups are used to help understand symmetry phenomena in molecular chemistry; and Poincaré groups can express the physical symmetry underlying special relativity.
qas: answers: symmetry
question: What shares an essential relationship to groups?

answers: Lie groups
question: What are the symmetry groups utilized as a part of the Standard Model particle physics?

answers: Point groups
question: What groups are are utilized toward comprehending symmetry wonders in chemistry? 

answers: Poincaré groups
question: What groups can express the physical symmetry hidden behind special relativity?

context: The concept of a group arose from the study of polynomial equations, starting with Évariste Galois in the 1830s. After contributions from other fields such as number theory and geometry, the group notion was generalized and firmly established around 1870. Modern group theory—an active mathematical discipline—studies groups in their own right.a[›] To explore groups, mathematicians have devised various notions to break groups into smaller, better-understandable pieces, such as subgroups, quotient groups and simple groups. In addition to their abstract properties, group theorists also study the different ways in which a group can be expressed concretely (its group representations), both from a theoretical and a computational point of view. A theory has been developed for finite groups, which culminated with the classification of finite simple groups announced in 1983.aa[›] Since the mid-1980s, geometric group theory, which studies finitely generated groups as geometric objects, has become a particularly active area in group theory.
qas: answers: the study of polynomial equations,
question: Where did the idea of a group come from?

answers: 1870
question: When was the group notion summed up and solidly settled?

answers: subgroups, quotient groups and simple groups.
question: What are smaller and easier to understand groups broken down into?

answers: 1983
question: When was announcement for the classification of finite simple groups?

context: The set G is called the underlying set of the group (G, •). Often the group's underlying set G is used as a short name for the group (G, •). Along the same lines, shorthand expressions such as "a subset of the group G" or "an element of group G" are used when what is actually meant is "a subset of the underlying set G of the group (G, •)" or "an element of the underlying set G of the group (G, •)". Usually, it is clear from the context whether a symbol like G refers to a group or to an underlying set.
qas: answers: The set
question: What is known as underlying set of the group?

answers: the group's underlying set
question: What is often utilized as a short name for the group?

answers: shorthand expressions
question: What expressions are utilized when is really implied to be a longer expression?

context: These symmetries are represented by functions. Each of these functions sends a point in the square to the corresponding point under the symmetry. For example, r1 sends a point to its rotation 90° clockwise around the square's center, and fh sends a point to its reflection across the square's vertical middle line. Composing two of these symmetry functions gives another symmetry function. These symmetries determine a group called the dihedral group of degree 4 and denoted D4. The underlying set of the group is the above set of symmetry functions, and the group operation is function composition. Two symmetries are combined by composing them as functions, that is, applying the first one to the square, and the second one to the result of the first application. The result of performing first a and then b is written symbolically from right to left as
qas: answers: functions
question: What sends a point in the square to the relating point under the symmetry? 

answers: Composing two of these symmetry functions
question:  What gives another symmetry function?

answers: the dihedral group
question: What group includes the symmetries of degree 4 and denoted D4?

context: The modern concept of an abstract group developed out of several fields of mathematics. The original motivation for group theory was the quest for solutions of polynomial equations of degree higher than 4. The 19th-century French mathematician Évariste Galois, extending prior work of Paolo Ruffini and Joseph-Louis Lagrange, gave a criterion for the solvability of a particular polynomial equation in terms of the symmetry group of its roots (solutions). The elements of such a Galois group correspond to certain permutations of the roots. At first, Galois' ideas were rejected by his contemporaries, and published only posthumously. More general permutation groups were investigated in particular by Augustin Louis Cauchy. Arthur Cayley's On the theory of groups, as depending on the symbolic equation θn = 1 (1854) gives the first abstract definition of a finite group.
qas: answers: Évariste Galois
question:  Which French mathematician expanded on earlier work of Paolo Ruffini and Joseph-Louis Lagrange?

answers: group theory
question: The journey for answers to polynomial equations of degree higher than 4 was the original motivation for what theory?

answers: Arthur Cayley
question:  Who developed a theory giving the first abstract definition of a finite group?

answers: abstract group
question: What modern concept was created from many fields of mathematics?

context: The convergence of these various sources into a uniform theory of groups started with Camille Jordan's Traité des substitutions et des équations algébriques (1870). Walther von Dyck (1882) introduced the idea of specifying a group by means of generators and relations, and was also the first to give an axiomatic definition of an "abstract group", in the terminology of the time. As of the 20th century, groups gained wide recognition by the pioneering work of Ferdinand Georg Frobenius and William Burnside, who worked on representation theory of finite groups, Richard Brauer's modular representation theory and Issai Schur's papers. The theory of Lie groups, and more generally locally compact groups was studied by Hermann Weyl, Élie Cartan and many others. Its algebraic counterpart, the theory of algebraic groups, was first shaped by Claude Chevalley (from the late 1930s) and later by the work of Armand Borel and Jacques Tits.
qas: answers: 1870
question: When did the uniform theory of groups develop from different sources? 

answers: Walther von Dyck
question: Who presented a method for specifying a group by means of generators and relations?

answers: The theory of Lie groups
question: What theory did Hermann Weyl study in addition to locally compact groups?

answers: Claude Chevalley
question: Who initially molded the theory of algebraic groups?

context: 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[›]
qas: answers: The University of Chicago
question: Where did the group of theorists first meet?

answers: 1960–61
question: What university class year belonged to the group of theorists?

answers: all finite simple groups
question: What did the group of theorists classify in 1982?

answers: length of proof and number of researchers.
question: What two factors describe the large size of the project?

context: 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.
qas: answers: structural concepts
question: What ideas are used to understand groups beyond symbols?

answers: constructions related to groups
question: What has to be compatible with the group operation?

answers: group homomorphisms
question: What concept describes groups that can be related to each other via functions?

context: 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.
qas: answers: isomorphic
question: What are two groups called if they include homomorphisms?

answers: isomorphic groups
question: What similar element do isomorphic groups carry?

answers: applying a to the first equality yields the second
question: How can showing the second equality yields the first prove the concept of isomorphic groups?

context: 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[›]
qas: answers: a rotation
question: What is composed of two rotations?

answers: inverse
question: What rotation can a rotation be reversed by?

answers: The subgroup test
question: What essential condition must be met for a subset of a group to be a subgroup?

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

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

answers: Cosets
question: What number sets are used to show how subgroups can be seen as translations of the larger group?

context: 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[›]
qas: answers: coset addition
question: What is another term for coset multiplication?

answers: coset multiplication
question: What group operation is taken by the set from the original group?

answers: quotient group
question: What group does the inverse of gN include?

context: 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
qas: answers: is the quotient of the free group over the generators of the group
question: What groups can be combined to describe every group?

answers: any group
question: What group can be the quotient of the free group over the generators of the group?

answers: every symmetry of the square
question: What is a finite composition of two symmetries or their inverses?

context: 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.
qas: answers: injective map
question: What map shows the relation between sub and quotient groups?

answers: surjective maps
question: What are the opposites of injective maps?

answers: canonical map
question: What is an example of a surjective map?

answers: the first isomorphism theorem
question: What theory address the phenomenon of homomorphisms being neither injective nor surjective?

context: 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.
qas: answers: Mathematical objects
question: What are usually analyzed by associating groups to them and studying the elements of the corresponding groups?

answers: Henri Poincaré
question: Who founded algebraic topology?

answers: the fundamental group
question: What did Henri Poincaré introduce when he established algebraic topology?

answers: proximity and continuity
question: What topological properties translate into properties of groups?

answers: loops
question: What represents elements of the fundamental group?

context: 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,
qas: answers: the modulus
question: What positive integer is used to divide the sum of two positive integers in modular mathematics?

answers: the remainder of that division
question: What results from modular addition?

answers: a clock
question: What type of device can be use to demonstrate modular addition?

context: 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
qas: answers: modulo p
question: What term describes the group of integers related to a prime number?

answers: 1 to p − 1
question: What integers are included in modulo p?

answers: four group elements
question: How many group elements exist if p=5?

context: 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.
qas: answers: cyclic
question: What is a group labeled when the element 1 is primitive?

answers: Any cyclic group with n elements
question: What group is isomorphic to cyclic groups?

answers: the group of n-th complex roots of unity
question: What example of cyclic group satisfies the express of zn = 1?

context: 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.
qas: answers: Symmetry groups
question: What are groups consisting of symmetries of given arithmetic concepts?

answers: geometric nature
question: What term describes the introductory symmetry group of the square?

answers: group theory
question: What can regarded as the study of symmetry?

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

answers: group action
question: The group pattern is connected to the structure of the target by what behavior?

context: 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.
qas: answers: group theory
question: What aids in predicting changes of physical traits?

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

answers: Curie temperature
question: What temperature causes the change of ferroelectric materials?

answers: soft phonon mode
question: What term describes the vibrational lattice mode that turns to 0 frequency at the change?

context: 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.
qas: answers: Mathieu groups
question: What are examples of finite symmetry groups used in coding theory?

answers: coding theory
question: What is used for error correction of transferred data?

answers: differential Galois theory
question: What describes functions having antiderivatives of a prescribed form?

answers: invariant theory
question: What concept investigates geometric elements that stay stable under group action?

context: 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.
qas: answers: Matrix groups
question: What groups combine matrices with matrix multiplication?

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

answers: matrix groups
question: What term describes subgroups of the general linear group?

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

answers: Euler angles
question: Rotation matrix groups are utilized in computer graphics with what concept?

context: 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.
qas: answers: degree 5 and higher
question: What degree does not include simple formulas for cubic and quatric equations?

answers: Abstract properties of Galois groups
question: What concept is associated with the solvability of polynomials?

answers: radicals
question: What are used to express the solutions of polynomials?

context: 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.
qas: answers: a finite number of elements
question: What does a finite group include?

answers: the order of the group
question: What is the number of elements in a group named?

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

answers: S3
question: What can be described as the group of symmetries of an equilateral triangle?

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.
qas: answers: complete classification
question: What level of finality do mathematicians try to reach with math concepts? 

answers: finite groups
question: What area of classification leads to troublesome arithmetic?

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

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

answers: The Jordan–Hölder theorem
question: What describes finite simple groups as the building pieces for all finite 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:
qas: answers: continuous functions
question: What group operations must occur for group law and topology to integrate well?

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

answers: Haar measures
question: What variables do locally compact groups share that can be studied by harmonic analysis?

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.
qas: answers: adele rings and adelic algebraic groups
question: What concepts are fundamental to number theory?

answers: the absolute Galois group
question: What group uses infinite field extensions with topology?

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

answers: 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?

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.
qas: answers: Lie groups
question: What concept is of basic importance in modern physics?

answers: Noether's theorem
question: What connects continuous symmetries to conserved quantities?

answers: Rotation
question: What term describes the basic symmetries of the laws of mechanics?

answers: Lorentz transformations
question: What concept relates measurements of time and velocity of two observers in motion relative to each other?

answers: Poincaré group
question: What describe the complete symmetry group of Minkowski space including translations?

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.
qas: answers: monoid
question: What product is created if the requirement that every element has an inverse is eliminated?

answers: natural numbers N (including 0)
question: What type of numbers under addition form a monoid?

answers: the Grothendieck group
question: What method formally adds inverses to elements to any monoid?

answers: the binary operation
question: What can be replaced to simplify abstract algebra concepts?