Difference between revisions of "Finitely generated group"
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Latest revision as of 09:19, 22 September 2016
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.
By definition, 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.
Every quotient of a finitely generated group is finitely generated. A subgroup of a finitely generated group need not be finitely generated.
See also
- Element (mathematics)
- Finitely generated module
- Generating set of a group
- Group (mathematics)
- Presentation of a group
External links
- Finitely generated group @ Wikpedia
