Generating set of a group

In abstract algebra, a generating set of a group <math>G<math> is a subset S such that every element of G can be expressed as the product of finitely many elements of S and their inverses. Whilst this may appear to produce excessively long strings of terms, in fact, these strings often can be reduced.

More generally, if S is a subset of a group G, then <S>, the subgroup generated by <math>S<math>, is the smallest subgroup of G containing every element of S; equivalently, <S> is the subgroup of all elements of G that can be expressed as the finite product of elements in S and their inverses.

If G = <S>, then we say S generates G; and the elements in S are called generators.

If S is the empty set, then <S> is the trivial group {e}, since we consider the empty product to be the identity.

When there is only a single element x in S, <S>; is usually written as <x>. In this case, <x> generates the cyclic subgroup of the powers of x.

If S is finite, then a group G = <S> is called finitely generated. The structure of finitely generated abelian groups in particular is easily described. Many theorems that are true for finitely generated groups fail for groups in general.

Every finite group is finitely generated since <G> = G. The integers under addition are an example of an infinite group which is finitely generated by both <1> and <−1>, but the group of rationals under addition cannot be finitely generated. No uncountable group can be finitely generated.

Different subsets of the same group can be generating subsets; for example, if p and q are integers with gcd(pq) = 1, then <{pq}> also generates the group of integers under addition.

The most general group generated by a set S is the group freely generated by S. Every group generated by S is isomorphic to a factor group of this group; a feature which is utilized in the expression of a group's presentation.

An interesting companion topic is that of non-generators. An element x of the group G is a non-generator if every set S containing x that generates G, still generates G when x is removed from S. In the integers with addition, the only non-generator is 0. The set of all non-generators forms a subgroup of G, the Frattini subgroup.

See also: Presentation of a group



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