Finitely generated group

The dihedral group of order 8 requires two generators, as represented by this cycle diagram.

In algebra, a finitely generated group is a group G that has some finite generating set S so that every element of G can be written as the product of finitely many elements of the finite set S and of inverses of such elements.^[1]

Every finite group is finitely generated, since S can be taken to be G itself. Every infinite finitely generated group must be countable but countable groups need not be finitely generated. The additive group of rational numbers Q is an example of a countable group that is not finitely generated.

A group that is generated by a single element is called cyclic. Every infinite cyclic group is isomorphic to the additive group of the integers Z.

Quotients and subgroups

Every quotient of a finitely generated group is finitely generated. However, a subgroup of a finitely generated group need not be finitely generated. For example, the commutator subgroup of the free group F₂ on two generators is not finitely generated. However, a subgroup of finite index in a finitely generated group is always finitely generated, and the Schreier index formula gives a bound on the number of generators required.^[2]

Finitely generated abelian groups

The six 6th complex roots of unity form a cyclic group under multiplication.

In a finitely generated abelian group with generators x₁, ..., x_n, every group element x can be written in the form

x = α₁⋅x₁ + α₂⋅x₂ + ... + α_n⋅x_s

with integers α₁, ..., α_n. (Every abelian group can be seen as a module over the ring of integers Z.)

The fundamental theorem of finitely generated abelian groups states that a finitely generated abelian group is the direct sum of a free abelian group of finite rank and a finite abelian group, each of which are unique up to isomorphism.

Related notions

The lattice of subgroups of a group satisfies the ascending chain condition if and only if all subgroups of the group are finitely generated. A group such that all its subgroups are finitely generated is called Noetherian.

The word problem for a finitely generated group is the decision problem whether two words in the generators of the group represent the same element. The word problem for a given finitely generated group is solvable if and only if the group can be embedded in every algebraically closed group.

A group is locally finite if every finitely generated subgroup is finite. Every locally finite group is periodic, i.e., every element has finite order. Conversely, every periodic abelian group is locally finite.^[3]

A group is locally cyclic if every finitely generated subgroup is cyclic. The additive group of the rational numbers Q is an example of a non-cyclic locally cyclic group.^[4] Every locally cyclic group is abelian.^[5] Every finitely-generated locally cyclic group is cyclic.

See also

• Finitely generated module
• Presentation of a group

Notes

1. Gregorac, Robert J. "A note on finitely generated groups" (PDF). American Mathematical Society. Retrieved 9 October 2015.
2. Rose 2012, p. 55
3. Rose 2012, p. 75
4. Rose 2012, p. 52
5. Rose 2012, p. 54

References

• Rose, John S. (2012) [unabridged and unaltered republication of a work first published by the Cambridge University Press, Cambridge, England, in 1978]. A Course on Group Theory. Dover Publications. ISBN 0-486-68194-7.