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:
“It is never the thing but the version of the thing:
The fragrance of the woman not her self,
Her self in her manner not the solid block,
The day in its color not perpending time,
Time in its weather, our most sovereign lord,
The weather in words and words in sounds of sound.”
—Wallace Stevens (1879–1955)
“Linguistically, and hence conceptually, the things in sharpest focus are the things that are public enough to be talked of publicly, common and conspicuous enough to be talked of often, and near enough to sense to be quickly identified and learned by name; it is to these that words apply first and foremost.”
—Willard Van Orman Quine (b. 1908)