Glossary of group theory

Please refer to group theory for a general description of the topic. See also list of group theory topics.

A group is a set together with an associative operation such that every element has an inverse.

Throughout the article, we use e to denote the identity element of a group.

Basic definitions

Order of a group. Order of a group (G,*) is the cardinal (i.e. the size) of G. A group with finit order is called a finite group.

Order of an element of a group. Suppose x∈G and there exists a positive integer m such that x^m=e, then the smallest possible m is called the order of x. The order of a finite group is divisible by the order of its every element.

Subgroup. A subset H of a group (G,*) which remains a group when the operation * is restricted to H is called a subgroup of G.

Given a set S of G. We denote by <S> to be the smallest subgroup of G containing S.

Normal subgroup. H is a normal subgroup G if for all g in G and h in H, g * h * g⁻¹ also belongs to H.

Both subgroups and normal subgroups of given group form a complete lattice under inclusion of subsets; this property and some related results are described by the lattice theorem[?].

Group homomorphism. These are functions f: (G,*) → (H,×) that have the special property that

f(a * b) = f(a) × f(b)

for any elements a and b of G.

Kernel of a group homomorphism. It is the preimage of the identity in the codomain of a group homomorphism. Every normal subgroup is the kernel of a group isomorphism and vice versa.

Group isomorphism. Group homomorphisms that have inverse functions. The inverse of an isomorphism, it turns out, must also be a homomorphism.

Isomorphic groups. Two groups are isomorphic if there exists an group isomorphism mapping from one to the other. Isomorphic groups can be thought as essentially the same, only with different labels on the individual elements. One of the fundamental problems of group theory is the classification of groups up to isomorphism.

Factor group, or quotient group. Given a group G and a normal subgroup N of G, the quotient group is the set G/N of left cosets {aN : a∈G'} together with the operation aN*bN=abN. The relationship between normal subgroups, homomorphisms, and factor groups is summed up in the fundamental theorem on homomorphisms.

Direct product, direct sums, and semidirect product of groups. They are ways to combining subgroups, please refer to the corresponding links for explanation.

Types of groups

Abelian group. A group (G,*) is abelian if * is commutative, i.e. gh=hg for all g,h∈G.

Finitely generated group[?]. If there exists a finite set S such that <S> = G, then G is said to be finitely generated.

p-group. If p is prime, then a p-group is just a group with order p^m for some m.

p-subgroup. A subgroup which is also p-group.

The study of p-subgroup is the central of the Sylow theorems.

Simple group. Simple groups are those groups with {e} and itself as the only normal subgroups. The name is misleading as its structure could be extremely complex. An example is the monster group, a group of order more than one million. Every finite group is built up from simple groups through the use of extensions, and in a celebrated huge theorem, classification of finite simple groups has been achieved.

The structure of any finite abelian group is relatively simple; every finite abelian group is the direct sum of cyclic p-groups. This can be extended to a complete classification of all finitely generated abelian groups, that is all abelian groups that are generated by a finite set.

The situation is much more complicated when trying to get a handle on the non-abelian groups.

Free group. Given any set A, one can define a multiplication of words as follows: (abb)*(bca)=abbbca. The free group generated by A is the smallest group containing this semigroup.

Every group (G,*) is basically a factor group of a free group generated by G. Please refer to presentation of a group for more explanation. One can then ask algorithmic questions about these presentations, such as:

• Do these two presentations specify isomorphic groups?; or
• Does this presentation specify the trivial group?

The general case of this is the word problem, and several of these questions are in fact unsolvable by any general algorithm.

General linear group. Denoted by GL(n), is the group of n-by-n invertible matrices.

Group representation. Not to be confused with the presentation of a group. A group representation is a homomorphism from a finite group to a general linear group. One basically tries to "represent" a given abstract group as a concrete group of invertible matrices which is much easier to study.