Glossary of Group Theory - Basic Definitions

Basic Definitions

A subset H ⊂ G is a subgroup if the restriction of • to H is a group operation on H. It is called normal, if left and right cosets agree, i.e. gH = Hg for all g in G. Normal subgroups play a distinguished role by virtue of the fact that the collection of cosets of a normal subgroup N in a group G naturally inherits a group structure, enabling the formation of the quotient group, usually denoted G/N (also called a factor group). The Butterfly lemma is a technical result on the lattice of subgroups of a group.

Given a subset S of a group G, the smallest subgroup of G containing S is called the subgroup generated by S. It is often denoted <S>.

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

Given any set A, one can define a group as the smallest group containing the free semigroup of A. This group consists of the finite strings called words that can be composed by elements from A and their inverses. Multiplication of strings is defined by concatenation, for instance

Every group G is basically a factor group of a free group generated by the set of its elements. This phenomenon is made formal with group presentations.

The direct product, direct sum, and semidirect product of groups glue several groups together, in different ways. The direct product of a family of groups G_i, for example, is the cartesian product of the sets underlying the various G_i, and the group operation is performed component-wise.

A group homomorphism is a map f : G → H between two groups that preserves the structure imposed by the operation, i.e.

f(a•b) = f(a) • f(b).

Bijective (in-, surjective) maps are isomorphisms of groups (mono-, epimorphisms, respectively). The kernel ker(f) is always a normal subgroup of the group. For f as above, the fundamental theorem on homomorphisms relates the structure of G and H, and of the kernel and image of the homomorphism, namely

G / ker(f) ≅ im(f).

One of the fundamental problems of group theory is the classification of groups up to isomorphism.

Groups together with group homomorphisms form a category.

In universal algebra, groups are generally treated as algebraic structures of the form (G, •, e, −1), i.e. the identity element e and the map that takes every element a of the group to its inverse a−1 are treated as integral parts of the formal definition of a group.