Finitely generated group

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

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

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.

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.

单词	Finitely Generated Group
释义	Finitely Generated Group 英语百科 Finitely generated group 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. 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.
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Finitely Generated Group

Finitely generated group