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:
“Those graceful acts,
Those thousand decencies, that daily flow
From all her words and actions, mixed with love
And sweet compliance, which declare unfeigned
Union of mind, or in us both one soul.”
—John Milton (1608–1674)
“And even my sense of identity was wrapped in a namelessness often hard to penetrate, as we have just seen I think. And so on for all the other things which made merry with my senses. Yes, even then, when already all was fading, waves and particles, there could be no things but nameless things, no names but thingless names. I say that now, but after all what do I know now about then, now when the icy words hail down upon me, the icy meanings, and the world dies too, foully named.”
—Samuel Beckett (1906–1989)