Karl Jones: Created page with "In algebra, a '''finitely generated group''' is a group G that has some finite generating set S so that every Eleme..."

2016-09-22T15:19:50Z

Created page with "In algebra, a '''finitely generated group''' is a group G that has some finite generating set S so that every Eleme..."

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In [[algebra]], a '''finitely generated group''' is a [[Group (mathematics)|group]] G that has some finite [[Generating set of a group|generating set]] S so that every [[Element (mathematics)|element]] of G can be written as the product of finitely many elements of the [[finite set]] S and of [[Inverse element|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 number|rational numbers]] Q is an example of a countable group that is not finitely generated.

Every [[Quotient group|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 ==

* [https://en.wikipedia.org/wiki/Finitely_generated_group Finitely generated group] @ Wikpedia

[[Category:Mathematics]]

Finitely generated group - Revision history

Karl Jones: Created page with "In algebra, a '''finitely generated group''' is a group G that has some finite generating set S so that every Eleme..."