Finitely-generated group

A group having a finite generating set . It thus consists of all products , , . If has elements, then is said to be a -generator group. Every generating set of a finitely-generated group contains a finite generating set. One-generator groups are said to be cyclic (they are isomorphic to either the additive group of integers, or the additive groups of residue classes of integers modulo , ).

The set of isomorphism classes of -generator groups has the cardinality of the continuum. Every countable group can be isomorphically imbedded in a -generator group; the imbedding group can be chosen to be simple and to be generated by an element of order 2 and one of order 3. Every countable -solvable group (cf. Solvable group) can be imbedded in a -generator -solvable group. Every subgroup of finite index in a finitely-generated group is finitely generated. A finitely-generated group has only finitely many subgroups of given finite index. A finitely-generated group can be infinite and periodic; in fact, for every natural number and every sufficiently large odd number there exists an infinite -generator group of exponent (see Burnside problem). A finitely-generated group can be isomorphic to a proper quotient group of itself; in this case it is called non-Hopfian (cf. Hopf group). There exist solvable non-Hopfian finitely-generated groups. A finitely-generated residually-finite group (see Residually-finite group) is Hopfian. Every finitely-generated group of matrices over a field is residually finite. There exist infinite finitely-generated, and even finitely-presented, groups that are simple (cf. Finitely-presented group).

References

[1]	M.I. Kargapolov, J.I. [Yu.I. Merzlyakov] Merzljakov, "Fundamentals of the theory of groups" , Springer (1979) (Translated from Russian)

Comments

For references see also Finitely-presented group.