Word (group Theory) - Words and Presentations

Words and Presentations

A subset S of a group G is called a generating set if every element of G can be represented by a word in S. If S is a generating set, a relation is a pair of words in S that represent the same element of G. These are usually written as equations:

A set of relations defines G if every relation in G follows logically from those in, using the axioms for a group. A presentation for G is a pair, where S is a generating set for G and is a defining set of relations.

For example, the Klein four-group can be defined by the presentation

Here 1 denotes the empty word, which represents the identity element.

When S is not a generating set for G, the set of elements represented by words in S is a subgroup of G. This is known as the subgroup of G generated by S, and is usually denoted . It is the smallest subgroup of G that contains the elements of S.