Generating Set of A Group - Finitely Generated Group

Finitely Generated Group

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. It has been proven that if a finite group is generated by a subset S, then each group element may be expressed as a word from the alphabet S of length less than or equal to the order of the group.

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(p, q) = 1, then {p, q} also generates the group of integers under addition (by Bézout's identity).

While it is true that every quotient of a finitely generated group is finitely generated (simply take the images of the generators in the quotient), a subgroup of a finitely generated group need not be finitely generated. For example, let G be the free group in two generators, x and y (which is clearly finitely generated, since G = <{x,y}>), and let S be the subset consisting of all elements of G of the form ynxy−n, for n a natural number. Since <S> is clearly isomorphic to the free group in countable generators, it cannot be finitely generated. However, every subgroup of a finitely generated abelian group is in itself finitely generated. In fact, more can be said: the class of all finitely generated groups is closed under extensions. To see this, take a generating set for the (finitely generated) normal subgroup and quotient: then the generators for the normal subgroup, together with preimages of the generators for the quotient, generate the group.