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.
Read more about this topic: Word (group Theory)
Famous quotes containing the words words and and/or words:
“‘A thing is called by a certain name because it instantiates a certain universal’ is obviously circular when particularized, but it looks imposing when left in this general form. And it looks imposing in this general form largely because of the inveterate philosophical habit of treating the shadows cast by words and sentences as if they were separately identifiable. Universals, like facts and propositions, are such shadows.”
—David Pears (b. 1921)
“The words which express our faith and piety are not definite; yet they are significant and fragrant like frankincense to superior natures.”
—Henry David Thoreau (1817–1862)