Misplaced Pages

This is an old revision of this page, as edited by TimothyRias (talk | contribs) at 12:47, 7 April 2008 (→Associativity). The present address (URL) is a permanent link to this revision, which may differ significantly from the current revision.

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. The branch of algebra that studies groups is called group theory. Group theory has extensive applications in mathematics, science, and engineering. Many algebraic structures such as fields and vector spaces may be defined concisely in terms of groups, and group theory provides an important tool for studying symmetry, since the symmetries of any object form a group. Groups are thus essential abstractions in branches of physics involving symmetry principles, such as relativity, quantum mechanics, and particle physics. Furthermore, their ability to represent geometric transformations finds applications in chemistry, computer graphics, and other fields.

Many structures investigated in mathematics turn out to be groups. These include familiar number systems, such as the integers, the rational numbers, the real numbers, and the complex numbers under addition, as well as the non-zero rationals, reals, and complex numbers under multiplication. Other important examples are the group of non-singular matrices under multiplication, and the group of invertible functions under composition. Group theory allows for the properties of such structures to be investigated in a general setting.

This article covers only the basic notions related to groups. More advanced facets, applications and history of group theory are covered in group theory.

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 an identity element.

Note that strictly speaking the first axiom (closure) is already implied by the condition that • be a binary operation on G. Many authors therefore omit this axiom.

Example

r0 (keeping it as is) r1 (rotation by 90°) r2 (rotation by 180°) r3 (left rotation by 90°) fv (vertical flip) fh (horizontal flip) f1 (diagonal flip) f2 (counterdiagonal flip)

group table • r0 r1 r2 r3 fv fh f1 f2 r0 r0 r1 r2 r3 fv fh f1 f2 r1 r1 r2 r3 r0 f2 f1 fv fh r2 r2 r3 r0 r1 fh fv f1 f2 r3 r3 r0 r1 r2 f1 f2 fh fv fv fv f1 fh f2 r0 r2 r1 r3 fh fh f2 fv f1 r2 r0 r3 r1 f1 f1 fh f2 fv r3 r1 r0 r2 f2 f2 fv f1 fh r1 r3 r2 r0

To clarify the group axioms we consider the group of symmetries of the square. The elements of the group will be operations that keep the shape of the square unchanged. (In the images, the vertices are colored only to visualize the operations). We have:

• Three rotations r₁, r₂ and r₃ (rotating the square by 90°, 180°, and 270° respectively).
• Reflections into the vertical and horizontal middle line (f_h and f_v), or into the two diagonals (f₁ and f₂).
• Finally, the identity operation r₀ leaving everything unchanged is also a symmetry.

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

   b • a ("perform the symmetry b after performing the symmetry a")

   For example, rotating by 90° left (r₃) and then flipping vertically (f_v) equals a reflection along the counterdiagonal (f₂). Using the above symbols, we have:

   f_v • r₃ = f₂ (highlighted in blue in the group table).

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 (f₁ • f_v) • r₂ = f₁ • (f_v • r₂) using the group table at the right:

   (f₁ • f_v) • r₂ = r₃ • r₂ = r₁, which equals

   f₁ • (f_v • r₂) = f₁ • f_h = r₁.

3. The identity element is the symmetry r₀ leaving everything unchanged: for any symmetry a, performing r₀ after a (or a after r₀) equals a, in symbolic form:

   r₀ • a = a, and

   a • r₀ = a.

4. Inverse elements are fulfilling the purpose of undoing the operation of some element. In the symmetry group example, every symmetry can be undone: the identity r₀, the flips f_h, f_v, f₁, f₂ and the 180° rotation r₂ are their own inverse, because repeating them brings the square back to its original orientation. The 90° rotations r₃ and r₁ are 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 = r₀,

   r₃ • r₁ = r₁ • r₃ = r₀.

Given their existence, both identity element and inverse elements are unique, see the notations section below.

Basic concepts in group theory

Elementary group theory is concerned with basic facts that hold for all individual groups, as opposed for example to the more involved study of groups via their representations. Such basic facts can usually be proved by invoking the axioms a few times, such as:

• 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.

• (Socks and shoes) The inverse of a product is the product of the inverses in the opposite order: (a • b) = b • a. To prove this, we will demonstrate that (a • b) • (b • a) = e, which as mentioned below under Inverse 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 neutral element)
	=	e	(definition of inverse)

The next level of studying groups is to look closer at their structure. Describing groups can be accomplished by constructing new groups from old, via sub- and quotient groups and products, by comparing different groups using homomorphisms, using particular features such as commutativity or by decomposing a group into simpler parts.

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 r₀, 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, see below under Cyclic groups.

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

By counting cosets, one can show Lagrange's Theorem: for a finite group G the order of any (necessarily finite) subgroup H divides the order of G.

Quotient groups

Quotient groups form the counterpart to subgroups. 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 groupG/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)${\displaystyle ^{-1}}$=N(g${\displaystyle ^{-1}}$).

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.

Simple groups

If a group G is not the trivial group and its only normal subgroups are the trivial group and the group itself, it is called a simple group. Equivalently, a group with only the trivial group and the group itself as quotient groups is simple. The Jordan-Hölder theorem exhibits simple groups as the building blocks for all other groups, the problem of the classification of finite simple groups was not solved until 1982.

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.

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 of a homomorphism f is denoted ker f and is the set of elements in G which are mapped to the identity in H. That is, ker f={g in G : f(g)=1_H}. The kernel of a homomorphism 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. A useful fact concerning homomorphisms is that they are injective if and only if their kernel is trivial (i.e. ker f={1_G}).

Abelian groups

A group ${\displaystyle G}$ 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" comes from the Norwegian mathematician Niels Henrik Abel. The above example of symmetries of the square is non-abelian, because

r₁ • f_v = f₂ ≠ f₁ = f_v • r₁.

The center of a group is a 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

A particularly easy class of abelian groups are cyclic groups, the groups whose elements may be generated by successive composition of the group operation being 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. It may or may not be the case that the sequence

..., −2•a, −1•a, 0, a, 2•a, ...

is periodic. If so, the group is isomorphic to Z_n (also denoted Z/nZ), where n is the smallest such period, i.e. the smallest integer such that n•a=0. Otherwise the group is isomorphic to (Z, +). An example of the first kind is the group of n-th complex roots of unity.

In any group, the successive composition of the operation applied to an element of the group generates a cyclic subgroup. By Lagrange's theorem, the order of the cyclic subgroup divides the order of the group. Thus, if the order of a finite group is prime, all of its elements, except the identity, are primitive elements of the group.

Order of groups and elements

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 the order is not finite, then the group is an infinite group, denoted |G| = ∞.

The order of an element a in a group G 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.

The order of the above example group is 8, the order of r₁ is 4, because rotating four times by 90° is not changing anything. The order of the reflection elements f_v etc. is 2.

Direct products and sums of groups

Besides subgroups and quotient groups there are several related 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 product 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.

Finally, a common generalization of the direct product 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 dihedral group of order 8 described in the introduction is a semidirect product of N = Z/4Z (the subgroup consisting of rotations) with H = Z/2Z (generated by a reflection).

Notations and remarks

When defining groups, it is standard notation to use parentheses in defining the group and its operation. For example, (H, +) denotes the group formed by the set H with addition as group operation.

Groups can use different notation depending on the context and the group operation. 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 field operations makes these sets into a group. It is also correct to refer to a group by its set identifier, e.g. H or Z, or to define the group in set-builder notation, provided it is clear which group operation is intended.}

Similar considerations apply to joining subsets by group operations: if S is a subset of a group G and x an element of G, then, in multiplicative notation, xS is the set of all products {xs : s in S} (a so-called coset), and for two subsets S and T of G, ST means the subset {st : s in S, t in T}. If the group is written additively, the respective sets are denoted x + S and S + T.

Identity element

Using the identity element property, it can be shown that a group has exactly one identity element. Therefore one usually speaks of the identity: suppose both e and f are identity elements. Then, because f is a (right) identity element e • f = e, and because e is a (left) identity element e • f = f, whence e = f.

The inverse

The inverse of an element a can also be shown to be unique, and it is usually written a or −a, depending on the context. Suppose given an inverse l and another inverse r. Then

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

Moreover, if in a group we know only that b • a = e, then this 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.

Associativity

For a sequence of multiple factors in a given order, one can form a product in many different ways by inserting parentheses; however, by several applications of the associativity property, any two of these can be shown to be equal. For this reason the expression

a₁ • a₂ • ... • a_n

is unambiguous and parentheses are usually omitted in such expressions. As a consequence it is hardly ever necessary to explicitly invoke the associativity property.

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, one may separately require the existence of a left and right identity element, and similarly one may 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).

Examples of groups

The integers under addition

The integers under addition form what is probably the most familiar group. One can think of the group axioms as being modelled on the properties of the integers Z = {..., −4, −3, −2, −1, 0, 1, 2, 3, 4, ...} together with the group operation "+", which denotes, as usual, the addition. The axioms to be checked are:

• Closure: If a and b are integers then a + b is an integer.
• Associativity: If a, b, and c are integers, then (a + b) + c = a + (b + c).
• Identity element: 0 is an integer and for any integer a, 0 + a = a + 0 = a.
• Inverse elements: If a is an integer, then the integer −a satisfies the inverse rules: a + (−a) = (−a) + a = 0.

This group is also abelian because a + b = b + a.

If we extend this example further by considering the integers with both addition and multiplication, it forms a more complicated algebraic structure called a ring. (But note that the integers with multiplication are not a group.)

Multiplicative groups

The integers under multiplication

To begin with, we give a counterexample: the integers with the operation of multiplication, denoted by "•". According to general notation, this is denoted (Z, •). 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. We cannot choose b = 1/2 because 1/2 is not an integer. Since not every element of (Z, •) has a (multiplicative) inverse, (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.

The nonzero rational numbers

The field of fractions is the natural step to remedy this. Consider the set of 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, if we instead use 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 (we do not lose closure by 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.

The 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, so it remains to prove is that each element has an inverse: given an integer a not divisible by p, one has to find an integer b such that

a · b ≡ 1 (mod p).

This 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

If the number of elements of a group G is finite, then G itself is called a finite group. The above dihedral group of order 8 is an example. Two important classes 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.

Lie groups

Lie groups 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.

Since in physics continuous symmetries are linked to conserved quantities via Noether's theorem, Lie groups tend to appear in physical theories. The Poincaré group, for example, plays a pivotal role in the theory of special relativity.

The general linear group and matrix groups

The general linear group GL(n) consists of all invertible n by n matrices together with matrix multiplication. The subgroups of GL(n) 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 fact, most of the important Lie groups in physics may be described as matrix groups. In chemical fields, such as crystallography, groups of rotation matrices are used to describe molecular symmetries.

Galois groups

Galois groups are the historical origin of group theory. They stem from the question which polynomials of degree greater than four have solutions by radicals, i.e. expressions involving repeated root operations and addition and multiplication comparable to the formula

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

solving the quadratic equation ax+bx+c=0. By associating to a polynomial its Galois group and studying in a purely abstract manner the properties of this group, in particular whether it is solvable or not, one can decide which polynomials do have a solution by radicals. It also explains the existence and structure of the formulae solving cubic and quartic equations.

Automorphism groups

In any (small) category the set of all automorphisms of an object V forms a group, Aut(V), under composition. Such groups are called automorphism groups. Many of the examples listed above are in fact automorphism groups. For example, GL(2) is the group of automorphisms of the vector space R.

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 relation, the resulting algebraic structure is called a magma.

Additionally:

• 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.
• Supergroups and Hopf algebras are other generalizations, and so are heaps.
• Abelian groups form the prototype for the concept of an abelian category, which has applications to vector spaces and beyond.
• Formal group laws are certain formal power series which have properties much like a group operation.

In differential geometry, algebraic geometry, and topology, the group concept specializes to include groups with additional structure. Lie groups, algebraic groups and topological groups are examples of group objects: group-like structures sitting in a category other than the ordinary category of sets.

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 (3 ed.), New York: Wiley, ISBN 978-0-471-43334-7, MR2286236.
• Herstein, I. N. (1996), Abstract algebra (3 ed.), Upper Saddle River, NJ: Prentice Hall Inc., ISBN 978-0-13-374562-7, MR1375019.
• 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).

See also

• List of group theory topics
• List of small groups

External links

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

• Abstract algebra
• Group theory
• Symmetry