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:
“One can describe a landscape in many different words and sentences, but one would not normally cut up a picture of a landscape and rearrange it in different patterns in order to describe it in different ways. Because a photograph is not composed of discrete units strung out in a linear row of meaningful pieces, we do not understand it by looking at one element after another in a set sequence. The photograph is understood in one act of seeing; it is perceived in a gestalt.”
—Joshua Meyrowitz, U.S. educator, media critic. “The Blurring of Public and Private Behaviors,” No Sense of Place: The Impact of Electronic Media on Social Behavior, Oxford University Press (1985)
“In all sincerity, we offer to the loved ones of all innocent victims over the past 25 years, abject and true remorse. No words of ours will compensate for the intolerable suffering they have undergone during the conflict.”
—Combined Loyalist Military Command. New York Times, p. A12 (October 14, l994)