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Algebraic structure → Group theory Group theory
Basic notions Subgroup Normal subgroup Group action Quotient group (Semi-)direct product Direct sum Free product Wreath product Group homomorphisms kernel image simple finite infinite continuous multiplicative additive cyclic abelian dihedral nilpotent solvable Glossary of group theory List of group theory topics
Finite groups Cyclic group Zn Symmetric group Sn Alternating group An Dihedral group Dn Quaternion group Q Cauchy's theorem Lagrange's theorem Sylow theorems Hall's theorem p-group Elementary abelian group Frobenius group Schur multiplier Classification of finite simple groups cyclic alternating Lie type sporadic
Discrete groups Lattices Integers (${\displaystyle \mathbb {Z} }$) Free group Modular groups PSL(2, ${\displaystyle \mathbb {Z} }$) SL(2, ${\displaystyle \mathbb {Z} }$) Arithmetic group Lattice Hyperbolic group
Topological and Lie groups Solenoid Circle General linear GL(n) Special linear SL(n) Orthogonal O(n) Euclidean E(n) Special orthogonal SO(n) Unitary U(n) Special unitary SU(n) Symplectic Sp(n) G2 F4 E6 E7 E8 Lorentz Poincaré Conformal Diffeomorphism Loop Infinite dimensional Lie group O(∞) SU(∞) Sp(∞)
Algebraic groups Linear algebraic group Reductive group Abelian variety Elliptic curve
v t e

A group is one of the fundamental objects of study in the field of mathematics known as abstract algebra, and more specifically group theory.

Many structures investigated in mathematics turn out to be groups, including familiar number systems such as the integers, or the rational numbers under addition, as well as the non-zero rational numbers under multiplication. Other important examples are groups of matrices, permutation groups and groups of symmetries of geometrical objects. Almost all algebraic structures such as rings and vector spaces can be defined concisely in terms of groups. Both relaxing and strengthening the requirements of the group axioms yields interesting further structures.

The theory of groups allows for the properties of such structures to be investigated in a general and abstract setting. Beyond direct implications of the group axioms, basic techniques include studying groups related to a given one (such as sub- or quotient groups) or decomposing groups into simpler parts. A particularly ample theory has been developed for finite and for abelian groups.

Groups are extensively used outside mathematics. Their ability to represent geometric transformations finds applications in chemistry and computer graphics. Groups, in particular Lie groups, are essential abstractions in branches of physics involving symmetry principles, such as relativity, quantum mechanics, and other fields.

Definition and illustration

A group (G, •) is a set G with a binary operation • on G that satisfies the following four axioms:

1.	Closure.	For all a, b in G, the result of a • b is also in G.
2.	Associativity.	For all a, b and c in G, (a • b) • c = a • (b • c).
3.	Identity element.	There exists an element e in G such that for all a in G, e • a = a • e = a.
4.	Inverse element.	For each a in G, there exists an element b in G such that a • b = b • a = e, where e is the identity element.

First example: the integers

The first example of a group is the set of integers Z = {..., −4, −3, −2, −1, 0, 1, 2, 3, 4, ...}. Under the usual addition operation "+" they form what is probably the most familiar group. The group axioms can be thought of as being modelled on the properties of the integers together with the addition operation. The set of integers under addition is denoted by (Z, +). The abstract group axioms reduce to statements about numbers, in this case:

1.	Closure.	For any two integers a, b, the sum a + b is also an integer.
2.	Associativity.	For all integers a, b and c, (a + b) + c = a + (b + c).
3.	Identity element.	If a is any integer, then 0 + a = a + 0 = a. Thus 0 is the (additive) identity.
4.	Inverse element.	For each a in Z, b = −a is an integer and satisfies a + b = b + a = 0. Thus −a is the (additive) inverse of the integer a.

Worked example: a symmetry group

id (keeping it as is) r1 (rotation by 90°) r2 (rotation by 180°) r3 (left rotation by 90°) fv (vertical flip) fh (horizontal flip) fd (diagonal flip) fc (counterdiagonal flip) Elements of the symmetry group. The vertices are colored only to visualize the operations.

group table • id r1 r2 r3 fv fh fd fc id id r1 r2 r3 fv fh fd fc r1 r1 r2 r3 id fc fd fv fh r2 r2 r3 id r1 fh fv fd fc r3 r3 id r1 r2 fd fc fh fv fv fv fd fh fc id r2 r1 r3 fh fh fc fv fd r2 id r3 r1 fd fd fh fc fv r3 r1 id r2 fc fc fv fd fh r1 r3 r2 id The elements id, r1, r2, and r3 form a subgroup (highlighted in red).

Group theory and the notion of a group concern much more general entities than numbers. The following illustrates the meaning of the group axioms for the dihedral group of symmetries of the square. The elements of the group are operations which keep the shape of the square unchanged. The operations are:

• Three rotations r₁, r₂ and r₃ (rotating the square by 90°, 180°, and 270° respectively).
• Reflections about the vertical and horizontal middle line (f_h and f_v), or through the two diagonals (f_d and f_c).
• The identity operation id leaving everything unchanged.

In this example group, the axioms can be understood as follows:

1. The closure axiom demands that any two symmetries can be composed. This is indeed the case—for any two symmetries a and b, we can first perform a and then b and the result will still be a symmetry, written symbolically from right to left ("perform the symmetry b after performing the symmetry a") as:

   b • a.

   For example, rotating by 90° left (r₃) and then flipping horizontally (f_h) is the same as performing a reflection along the diagonal (f_d). Using the above symbols, we have (highlighted in blue in the group table):

   f_h • r₃ = f_d.

2. The associativity constraint is the natural axiom to impose in order to make composing more than two symmetries well-behaved: given three elements a, b and c of G, there are two possible ways of computing "a after b after c". The requirement:

   (a • b) • c = a • (b • c)

   means that composing a after b, and calling this symmetry x, then x after c is the same as a after y, where y in turn is applying b after c. For example, we check (f_d • f_v) • r₂ = f_d • (f_v • r₂) using the group table at the right:

   (f_d • f_v) • r₂ = r₃ • r₂ = r₁, which equals

   f_d • (f_v • r₂) = f_d • f_h = r₁.

3. The identity element is the symmetry id leaving everything unchanged: for any symmetry a, performing id after a (or a after id) equals a, in symbolic form:

   id • a = a, and

   a • id = a.

4. An inverse element undoes the operation of some other element. In the symmetry group example, every symmetry can be undone: each of the identity id, the flips f_h, f_v, f_d, f_c and the 180° rotation r₂ is its own inverse, because performing each one twice brings the square back to its original orientation. Each of the 90° rotations r₃ and r₁ is each other's inverse, because rotating one way and then by the same angle the other way leaves the square unchanged. In symbols, for example:

   f_h • f_h = id,

   r₃ • r₁ = r₁ • r₃ = id.

First consequences of the group axioms

Elementary group theory is concerned with basic facts about general groups, as opposed for example to the more involved study of groups via their representations. These facts are usually direct consequences of the group definition — obtained by invoking the axioms a few times — and are often used in group theory without explicit reference to the corresponding statement.

For example, repeated applications of the associativity axiom show that the unambiguity of

a • b • c = (a • b) • c = a • (b • c)

generalizes to more than three factors. Therefore parentheses are usually omitted in such expressions.

Uniqueness of identity element and inverses

Though the uniqueness of the identity is not required by the group axioms, it is a consequence of them. Therefore it is customary to speak of the identity, and the inverse of a.

The following proof of this fact shows the flavor of elementary group theory: suppose both e and f are identity elements. Then

e = e • f = f,

because e is a (left) identity element and f is a (right) identity element. Hence the two identities necessarily agree. Similarly, suppose given two inverses l and r of a fixed element a. Then

l = l • e = l • (a • r) = (l • a) • r = e • r = r.

Moreover, if in a group knowing only that b • a = e suffices to conclude that b is the inverse element of a (since a two-sided inverse of a is guaranteed to exist, and then b must be equal to it). Similarly a •  b = e suffices for the same conclusion.

The inverse of a product is the product of the inverses in the opposite order: (a • b) = b • a. The identity (a • b) • (b • a) = e then suffices to prove that b • a is the inverse of a • b.

(a • b) • (b • a)	=	((a • b) • b ) • a	(associativity)
	=	(a • (b • b)) • a	(associativity)
	=	(a • e) • a	(definition of inverse)
	=	a • a	(definition of identity element)
	=	e	(definition of inverse)

Division

In groups, it is possible to perform division: given elements a and b of the group G, there is exactly one solution x in G to the equation x • a = b. In fact, right multiplication of the equation by a gives the solution x = b • a. Similarly there is exactly one solution y in G to the equation a • y = b, namely y = a • b.

Variants of the definition

Some definitions of a group use seemingly weaker conditions for identity and inverse elements. Instead of requiring a two-sided identity element, the axioms may separately require the existence of a left and right identity element, and similarly they may only separately require the existence of a left and right inverse elements. In both cases the left and right elements can be shown to be the same (and each is unique).

Strictly speaking the closure axiom is already implied by the condition that • be a binary operation on G. Many authors therefore omit this axiom.

In universal algebra, groups are generally treated as algebraic structures of the form (G, •, e, ), i.e. the identity element e and the map that takes every element a of the group to its inverse a are treated as integral parts of the formal definition of a group.

Notations

The notation for groups often depends on the context and the nature of the group operation. There is a tendency to denote abelian groups additively, whereas non-abelian groups are often written multiplicatively. In many situations, there is only one possible (or reasonable) group operation on a given set, therefore it is very common to drop the operation symbol and leave it to the reader to know the context and the group operation. For example the groups (Z_n, +) and (F_q*, ×), the multiplicative group of nonzero elements in the finite field F_q are commonly denoted Z_{n and Fq*, since only one of the two ring operations makes these sets into a group.}

Basic concepts in group theory

Algebraic structure → Group theory Group theory
Basic notions Subgroup Normal subgroup Group action Quotient group (Semi-)direct product Direct sum Free product Wreath product Group homomorphisms kernel image simple finite infinite continuous multiplicative additive cyclic abelian dihedral nilpotent solvable Glossary of group theory List of group theory topics
Finite groups Cyclic group Zn Symmetric group Sn Alternating group An Dihedral group Dn Quaternion group Q Cauchy's theorem Lagrange's theorem Sylow theorems Hall's theorem p-group Elementary abelian group Frobenius group Schur multiplier Classification of finite simple groups cyclic alternating Lie type sporadic
Discrete groups Lattices Integers (${\displaystyle \mathbb {Z} }$) Free group Modular groups PSL(2, ${\displaystyle \mathbb {Z} }$) SL(2, ${\displaystyle \mathbb {Z} }$) Arithmetic group Lattice Hyperbolic group
Topological and Lie groups Solenoid Circle General linear GL(n) Special linear SL(n) Orthogonal O(n) Euclidean E(n) Special orthogonal SO(n) Unitary U(n) Special unitary SU(n) Symplectic Sp(n) G2 F4 E6 E7 E8 Lorentz Poincaré Conformal Diffeomorphism Loop Infinite dimensional Lie group O(∞) SU(∞) Sp(∞)
Algebraic groups Linear algebraic group Reductive group Abelian variety Elliptic curve
v t e

The structure of groups can be understood by breaking them into pieces, combining them into larger groups, or by comparing them to other groups. The pieces are called subgroups and quotient groups. The larger groups are called direct products and semi-direct products. Groups are compared using homomorphisms. Some groups are singled out as particularly important building blocks: the cyclic, abelian, and simple groups. These basic concepts form the standard introduction to groups.

Subgroups

A subset H ⊂ G is called a subgroup if the restriction of • to H is a group operation on H. In other words, it is a group using the restriction of the operation defined on G. In the example above, the rotations constitute a subgroup, since a rotation composed with a rotation is still a rotation: in the group table, the intersections of rows and columns for id, r₁, r₂, and r₃, only contain those same elements (highlighted in red). It can be read off the group table above, and is indeed a general principle that knowing the subgroups of a group is important to understand the structure of the group in question.

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 gh ∈ H for all g, h ∈ H. The closure, under the group operation and inversion, of any nonempty subset of a group is a subgroup. The subset is said to generate the subgroup. For example, the powers of any element a and their inverses (that is, a = e, a, a, a, a, …, a, a, a, a, …) always form a subgroup of the larger group, the so-called cyclic subgroup generated by a.

A subgroup H defines a set of left and right cosets. Given an arbitrary element g in G, the left and right coset of H containing g are

gH = {gh, h ∈ H} and Hg = {hg, h ∈ H}, respectively.

The set of left cosets of H forms a partition of the elements of G; that is, two left cosets are either equal or have an empty intersection. The same holds true of the right cosets of H. Left and right cosets of H may or may not be equal. If it is the case that for all g in G, gH = Hg, then H is said to be a normal subgroup.

Quotient groups

Quotient groups, also known as factor groups, treat the cosets of a normal subgroup as a group. If N is a normal subgroup of G, its set of left cosets and right cosets are the same and one may speak simply of the set of cosets of N. In this case, the set of cosets of N may be equipped with an operation (sometimes called coset multiplication, or coset addition) to form a new group, called the quotient group G/N. The operation between the cosets behaves in the nicest way possible: (Ng) • (Nh) = N(gh) for all g and h in G. The coset N itself serves as the identity in this group, and the inverse of Ng in the quotient group is (Ng) = N(g).

Quotient 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 introductory dihedral group, for example, is presented by two generators r and f (for example, r = r₁, the right rotation and f = f_v the vertical (or any other) flip), together with the relations

r = f = (rf) = 1.

This way of describing a group can also be used to construct the Cayley graph, a graphical device showing certain features of discrete groups.

Group homomorphisms

If G and H are two groups, a group homomorphism f is a mapping f: G → H that preserves the structure of the groups in question. The structure of groups being determined by the group operation, this means the following: if g and k are any two elements in G, then

f(gk)=f(g)f(k).

This requirement ensures that f(1_G)=1_H, and also f(g)=f(g) for all g in G. The category of groups is the abstract framework containing groups and group homomorphisms.

Two groups G and H are called isomorphic if there exists a group homomorphism f between G and H which is both surjective (onto) and injective (one-to-one). The kernel and image of a homomorphism can be interpreted as measuring how close it is to being an isomorphism. The kernel ker f={g in G : f(g)=1_H} is the set of elements in G which are mapped to the identity in H. It is always a normal subgroup. The First Isomorphism Theorem states that the image of a group homomorphism, f(G) is isomorphic to the quotient group G/ker f.

Simple groups

A nontrivial group is called simple if its only normal subgroups are the trivial group and the group itself. Equivalently, a nontrivial group is simple if its only quotient groups are the trivial group and the group itself. The Jordan-Hölder theorem exhibits simple groups as the building blocks for all finite groups. The classification of finite simple groups was a major problem in group theory. Filling the gaps in the 1982 proof and simplifying it are areas of active research.

Abelian groups

A group G is said to be abelian, or commutative, if the operation satisfies the commutative law. That is, for all a and b in G, a • b = b • a. If not, the group is called non-abelian or non-commutative. The name "abelian" derives from the Norwegian mathematician Niels Henrik Abel. The group of symmetries of the square (discussed above) is non-abelian, because

r₁ • f_v = f_c ≠ f_d = f_v • r₁.

but there is an abelian subgroup {id, r₁, r₂, r₃} consisting of only the rotations.

The center of a group is the subgroup consisting of the elements which commute with every other element in the group. In a commutative group the center is the whole group; at the other extreme there are groups whose center is trivial, i.e. it consists only of the identity element.

Cyclic groups

Cyclic groups are those groups whose elements may be generated by successive compositions of the group operation applied to a single element of that group. An element with this property is called a generator or a primitive element of the group. Written additively, the group is therefore generated by the multiples t • a (the multiplicative notation would be a), where t ranges in Z. A cyclic group may or may not be finite. If it is finite, the group is isomorphic to Z_n (also denoted Z/nZ), where n is the smallest integer such that n • a = 0, for example the group of n-th complex roots of unity. Otherwise the group is isomorphic to (Z, +).

In any group, the successive composition of the operation applied to an element of the group generates a cyclic subgroup.

Products and sums of groups

In addition to subgroups and quotient groups there are further ways of constructing new groups from given ones: given two groups (G, *) and (H, •), their direct product is the set G×H together with the operation

(g₁,h₁)(g₂,h₂) :=(g₁*g₂,h₁•h₂).

This definition extends to products of any number of groups, finite or infinite, by using the Cartesian product. A variation of this construction is the direct sum, the subgroup of the product constituted by elements that have only a finite number of non-identity coordinates. If the family is finite the direct sum and the product are equivalent.

A further generalization of the direct product of two groups is the semidirect product. Given two groups N and H, it allows for the twisting of the group operation on the first factor by a group homomorphism φ : H → Aut(N): the semidirect product of N and H with respect to φ is the group (N × H, •), with • defined as

(n₁, h₁) • (n₂, h₂) = (n₁ φ(h₁) (n₂), h₁ h₂).

The group of symmetries of the square (described above) is a semidirect product of N = Z/4Z (the subgroup consisting of rotations) with H = Z/2Z (generated by a reflection).

Examples of groups

Integers under addition

The integers Z under addition form a group (described above). In addition to merely being a group, this group is also abelian because

a + b = b + a (commutativity of addition).

The integers are the basic building block for abelian groups, for example every torsion-free group contains (Z, +) as a subgroup.

Multiplicative groups

Many structures involving multiplication also form groups, however many such structures may well fail to be groups. An important counterexample is the integers, denoted by Z, with the operation of multiplication, denoted by "•". The pair (Z, •) is not a group. It satisfies the closure, associativity and identity axioms, but fails to have inverses: it is not true that whenever a is an integer, there is an integer b such that ab = ba = 1. For example, a = 2 is an integer, but the only solution to the equation ab = 1 in this case is b = 1/2, which is a rational number, but not an integer. Hence not every element of (Z, •) has a (multiplicative) inverse, so (Z, •) is not a group. It is, however, a commutative monoid, which is a similar structure to a group but does not require inverse elements. From another point of view, the integers, with both addition and multiplication form a more complicated algebraic structure called a ring.

Nonzero rational numbers

The natural remedy of the above situation are the rational numbers Q, the set of all fractions of integers a/b, where a and b are integers and b is nonzero, and the multiplication operation, again denoted by "•". Since the rational number 0 does not have a multiplicative inverse, (Q, •), like (Z, •), is not a group.

However, the set of all nonzero rational numbers Q \ {0}, then (Q \ {0}, •) does form an abelian group. Indeed, closure, associativity and identity element axioms are easy to check and follow from the properties of integers. The closure requirement still holds true after removing zero, because the product of two nonzero rationals is never zero. Finally, the inverse of a/b is b/a, therefore the axiom of the inverse element is satisfied.

Just as the integers form a ring, the rational numbers form the algebraic structure of a field, allowing the operations of addition, subtraction, multiplication and division.

Cyclic multiplicative groups

In (Q \ {0}, •), there are the cyclic subgroups

G = {a, n ∈ Z} ⊂ Q

where a is the n-th exponentiations of the primitive element a of that group. For example, if a is 2 then

G = {..., 2, 2, 2, 2, 2, ...} = {..., 0.25, 0.5, 1, 2, 4, ...}.

This group is an example of a free abelian group of rank one: the rank is one, because G is generated by one element (a or equivalently a) and the freeness refers to the fact that no relations between the powers of this generator occur. Therefore, G, is isomorphic to the (additive) group of integers (Z, +) above. This example shows that distinguishing between additive and multiplicative groups is merely a matter of notation – group theory treats groups from a purely abstract point of view, forgetting about the concrete nature of the group elements and the group operation.

Nonzero integers modulo a prime

The nonzero classes of integers modulo p, a prime number, form a group under multiplication called the multiplicative group of integers modulo p. The product of two integers neither of which is divisible by p is not divisible by p either (because p is prime), which shows that the indicated set of classes is closed under multiplication. Associativity is clear, and the class of 1 is the identity for multiplication. The inverse element axiom requires that given an integer a not divisible by p, there exists an integer b such that

a · b ≡ 1 (mod p).

It can be shown by inspecting the structure of the finite field F_p. Actually, this example is similar to (Q\{0}, •) above, because it turns out to be the group of nonzero elements in F_p.

Finite groups

The order of a group G, usually denoted by |G| or occasionally by o(G), is the number of elements in the set G. If it is finite, then G itself is called a finite group. Otherwise, the group is an infinite group. Via arithmetic geometry over finite fields, finite groups are much applied in cryptography. Two imporant classes of finite groups are the following:

• the cyclic (abelian) groups Z/nZ treated above. Any abelian finite group is a finite direct sum of groups of this kind, this is part of the fundamental theorem of finitely generated abelian groups.
• the symmetric group S_N: it is the group of permutations of N letters. For example, the symmetric group on 3 letters S₃ is the group consisting of all possible swaps of the three letters ABC, i.e. contains the elements ABC, ACB, ..., up to CBA, in total 6 (or 3 factorial) elements. Parallel to the group of symmetries of the square above, S₃ can also be interpreted as the group of symmetries of an equilateral triangle.

Cayley's theorem states that any finite (not necessarily abelian) group can be expressed as a subgroup of a symmetric group S_N.

The order of an element a in a group G, often denoted o(a), is the least positive integer n such that a = e, where a represents ${\displaystyle \underbrace {a\cdot \ldots \cdot a} _{n}}$, i.e. application of the operation • to n copies of the value a. (If • represents multiplication, then a corresponds to the n power of a.) If no such n exists, then the order of a is said to be infinity. The order of an element is the same as the order of the cyclic subgroup generated by this element.

More sophisticated counting techniques, for example counting cosets, yield more precise statements about finite groups: Lagrange's Theorem states that for a finite group G the order of any (necessarily finite) subgroup H divides the order of G. A partial converse is given by the Sylow theorems.

The dihedral group (discussed above) is a finite group of order 8. The order of r₁ is 4, because rotating four times by 90° does not change anything. The order of the reflection elements f_v etc. is 2. Both orders divide 8, as predicted by Lagrange's Theorem.

Symmetry groups

Symmetry groups are groups consisting of symmetries of given geometrical objects. The introductory dihedral group is an example for a symmetry group. Symmetries greatly simplify the study of geometrical objects. This remark is formalized and exploited using the notion of group actions, a concept much used in geometric invariant theory. This way, abstract algebra and geometry are enriching each other: the methods of abstract algebra are applied to get refined information about geometrical objects whereas geometry may serve as an intuiton for advances in abstract algebra. Symmetry groups play a central rôle in modern physical fields such as gauge theory. Finite symmetry groups such as the Mathieu groups are used in coding theory, which is in turn applied in error correction of transmitted data, or in CD players. Every group is the symmetry group of some graph.

Lie groups

In many situations groups are endowed with an additional structure. Examples include algebraic and topological groups studied in algebraic geometry and topology and Lie groups. The latter are groups which also have a (compatible) manifold structure. Because of the manifold structure it is possible to consider continuous paths in the group. For this reason they are also referred to as continuous groups. Examples include the real numbers with addition, the nonzero complex numbers with multiplication, or the rotations of a circle with composition.

In physics continuous symmetries are linked to conserved quantities via Noether's theorem, so Lie groups play an important role in physical theories. The Poincaré group, for example, plays a pivotal role in special relativity and quantum field theory.

General linear group and matrix groups

Most of Lie groups important in physics may be described as groups of matrices together with matrix multiplication: the general linear group GL(n,K) consists of all invertible n by n matrices with entries in a fixed field K. From a category-theoretic point of view, this group is an example of an automorphism group, for it is the group of linear automorphisms of the vector space K.

The subgroups of GL(n,K) are referred to as matrix groups. The dihedral group example mentioned above can be viewed as a (very small) matrix group. Other important matrix groups include the special orthogonal group SO(n) which describes all possible rotations in n dimensions. In chemical fields, such as crystallography, groups of rotation matrices are used to describe molecular symmetries.

Galois groups

Galois groups are groups of substitutions of the roots of a polynomial and were historically important during the development of group theory. They stem from the question of determining which polynomials of degree greater than four have solutions by radicals. That is, which polynomial have all their solutions expressible using solely addition, multiplication, and n-th roots, where n is a positive integer. For example, the solutions of the quadratic equation ax+bx+c=0 may be expressed by radicals as

${\displaystyle x={\frac {-b\pm {\sqrt {b^{2}-4ac}}}{2a}}}$.

The abstract properties of Galois groups (in particular their solvablility) associated to polynomials give a criterion which polynomials do have all their solutions expressible by radicals. Galois theory also explains the existence and structure of the formulae solving cubic and quartic equations.

Generalizations

In abstract algebra, more general structures arise by relaxing some of the axioms defining a group:

• Eliminating the requirement that every element have an inverse, then the resulting algebraic structure is called a monoid.
• A monoid without an identity is called a semigroup.
• Alternatively, relaxing the requirement that the operation be associative while still requiring the possibility of division, the resulting algebraic structure is a loop.
• A loop without an identity is called a quasigroup.
• Finally, dropping all axioms for the binary operation, the resulting algebraic structure is called a magma.
• Groupoids, which are similar to groups except that the composition a • b need not be defined for all a and b, arise in the study of more involved kinds of symmetries, often in topological and analytical structures, e.g. the fundamental groupoid. Groupoids, in turn, are special sorts of categories.

Additionally:

• Generalizing from binary to ternary operations leads to heaps.
• The category of abelian groups form the prototype for the concept of an abelian category, which provides the general setting for homological algebra.
• Groups may be viewed as a special case of a Hopf algebra, deforming a group in this larger context leads to the concept of a quantum group.
• Formal group laws are certain formal power series which have properties much like a group operation.

See also

• List of group theory topics
• List of small groups

Notes

1. Herstein 1975, section 2, p. 26.
2. Hall 1967, section 1.1., p. 1: "The idea of a group is one which pervades the whole of mathematics both pure and applied."
3. Herstein 1975, section 2.1., p. 27
4. Herstein 1975, section 2.6., p. 54
5. Fulton & Harris 1991
6. Ledermann 1953, section 1.2., pp. 4–5
7. Lang 2005, section II.1., p. 17
8. see for example Artin, Emil, Algebra I {{citation}}: Text "Vorlesungsnotizen, Universität Hamburg" ignored (help), Definition 1.7., p. 4.
9. such as Lang 2002
10. Artin 1991, section 2.1.
11. In the context of commutative algebra, Z_n is usually denoted Z/n Z or Z/n, see for example Eisenbud 1995
12. see Lang's and Herstein's books below
13. Lang 2005, section II.1., p. 19
14. Lang 2005, section II.4., p. 41
15. Lang 2002, section I.2., p. 12
16. Lang 2005, section II.4., p. 45
17. See universal property
18. Lang 2002, section I.2., p. 9
19. Lang 2005, section II.3., p. 34
20. The converse statement is also true: any normal subgroup N is the kernel of the canonical map G → G/N, see Lang 2005, section II.4., p. 45
21. Michler 2006, Carter 1989
22. Lang 2002, section I.3., p. 22
23. Aschbacher 2004
24. Lang 2002, section I.1., p. 4
25. Lang 2002, section I.5., p. 26, 29.
26. Lang 2005, section II.1., p. 22
27. Lang 2005, section II.2., p. 26
28. Lang 2005, Example 11, section II.1., p. 22
29. Lang 2002, section I.2.
30. Namely the cyclic subgroup generated by an arbitrary non-identity element.
31. Elements which do have multiplicative inverses are called units, see Lang 2002, section II.1., p. 84
32. The same is true for any field F instead of Q. See Lang 2005, section III.1., p. 86.
33. This example is from Lang 2005, section II.1., p. 22
34. Lang 2005, chapter VII
35. Kurzweil & Stellmacher 2004
36. For example, the Diffie-Hellman protocol uses the discrete logarithm.
37. Weyl 1952
38. See, for example, hyperbolic groups, which employ ideas and methods of hyperbolic geometry.
39. Welsh 1989
40. Frucht 1939
41. Warner 1983
42. Borel 1991
43. Robinson 1996, p. viii
44. Artin 1998
45. Lang 2002, theorem VI.7.2., p. 292
46. Mac Lane 1998
47. Denecke & Wismath 2002
48. Romanowska & Smith 2002
49. Kassel 1994
50. Fröhlich 1968

References

• Important historical publications in group theory.
• Artin, Michael (1991), Algebra, Prentice Hall, ISBN 978-0-89871-510-1, Chapter 2 contains a undergraduate-level exposition of the notions covered in this article.
• Devlin, Keith (2000), The Language of Mathematics : Making the Invisible Visible, Owl Books, ISBN 978-0-8050-7254-9, Chapter 5 provides a layman-accessible explanation of groups.
• Dummit, David S.; Foote, Richard M. (2004), Abstract algebra (3rd ed.), New York: Wiley, ISBN 978-0-471-43334-7, MR2286236.
• Fulton, William; Harris, Joe (1991), Representation Theory, A First Course, Graduate Texts in Mathematics, vol. 129, Berlin, New York: Springer-Verlag, ISBN 978-0-387-97495-8, MR1153249
• Hall, G. G. (1967), Applied group theory, American Elsevier Publishing Co., Inc., New York, MR0219593, an elementary introduction
• Herstein, Israel Nathan (1996), Abstract algebra (3rd ed.), Upper Saddle River, NJ: Prentice Hall Inc., ISBN 978-0-13-374562-7, MR1375019.
• Herstein, Israel Nathan (1975), Topics in algebra (2nd ed.), Lexington, Mass.: Xerox College Publishing, MR0356988
• Lang, Serge (2002), Algebra, Graduate Texts in Mathematics, vol. 211, Berlin, New York, ISBN 978-0-387-95385-4, MR1878556{{citation}}: CS1 maint: location missing publisher (link)
• Lang, Serge (2005), Undergraduate Algebra (3rd ed.), Berlin, New York: Springer-Verlag, ISBN 978-0-387-22025-3
• Ledermann, Walter (1953), Introduction to the theory of finite groups, Oliver and Boyd, Edinburgh and London, MR0054593
• Robinson, Derek John Scott (1996), A course in the theory of groups, Berlin, New York: Springer-Verlag, ISBN 978-0-387-94461-6

Further reading

• Artin, Emil (1998), Galois Theory, New York: Dover Publications, ISBN 978-0-486-62342-9
• Aschbacher, Michael (2004), "The Status of the Classification of the Finite Simple Groups" (PDF), Notices of the American Mathematical Society, 51 (7): 736–740, ISSN 0002-9920. See also the references in the classification article.
• Borel, Armand (1991), Linear algebraic groups, Graduate Texts in Mathematics, vol. 126 (2nd ed.), Berlin, New York: Springer-Verlag, ISBN 978-0-387-97370-8, MR1102012
• Carter, Roger W. (1989), Simple groups of lie type, New York: John Wiley & Sons, ISBN 978-0-471-50683-6
• Denecke, Klaus; Wismath, Shelly L. (2002), Universal algebra and applications in theoretical computer science, London: CRC Press, ISBN 978-1-58488-254-1
• Eisenbud, David (1995), Commutative algebra, Graduate Texts in Mathematics, vol. 150, Berlin, New York: Springer-Verlag, ISBN 978-0-387-94268-1; 978-0-387-94269-8, MR1322960 {{citation}}: Check |isbn= value: invalid character (help)
• Fröhlich, Albrecht (1968), Formal groups, Lecture notes in mathematics, vol. 74, Berlin, New York: Springer-Verlag
• Frucht, R. (1939), "Herstellung von Graphen mit vorgegebener abstrakter Gruppe", Compositio Mathematica, 6: 239–50, ISSN 0010-437X
• Kassel, Christian (1994), Quantum Groups, Springer, ISBN 978-0387943701
• Kurzweil, Hans; Stellmacher, Bernd (2004), The theory of finite groups, Universitext, Berlin, New York: Springer-Verlag, ISBN 978-0-387-40510-0, MR2014408
• Mac Lane, Saunders (1998), Categories for the Working Mathematician (2nd ed.), Berlin, New York: Springer-Verlag, ISBN 978-0-387-98403-2
• Michler, Gerhard (2006), Theory of finite simple groups, Cambridge University Press, ISBN 978-0-521-86625-5
• Romanowska, A.B.; Smith, J.D.H. (2002), Modes, World Scientific, ISBN 9789810249427
• Warner, Frank (1983), Foundations of Differentiable Manifolds and Lie Groups, Berlin, New York: Springer-Verlag, ISBN 978-0-387-90894-6
• Welsh, Dominic (1989), Codes and cryptography, Oxford: Clarendon Press, ISBN 978-0-19-853287-3
• Weyl, Hermann (1952), Symmetry, Princeton University Press, ISBN 978-0-691-02374-8

External links

• Weisstein, Eric W. "Group". MathWorld.
• Group at PlanetMath.

• Abstract algebra
• Group theory
• Symmetry