# virtually abelian group

A group $G$ is virtually abelian (or abelian-by-finite) if it has an abelian subgroup (http://planetmath.org/Subgroup) of finite index (http://planetmath.org/Coset).

More generally, let $\chi$ be a property of groups. A group $G$ is virtually $\chi$ if it has a subgroup of finite index with the property $\chi$. A group $G$ is $\chi$-by-finite if it has a normal subgroup of finite index with the property $\chi$. Note that every $\chi$-by-finite group is virtually $\chi$, and the converse also holds if the property $\chi$ is inherited by subgroups.

These notions are obviously only of relevance to infinite groups, as all finite groups are virtually trivial (and trivial-by-finite).

 Title virtually abelian group Canonical name VirtuallyAbelianGroup Date of creation 2013-03-22 14:35:58 Last modified on 2013-03-22 14:35:58 Owner yark (2760) Last modified by yark (2760) Numerical id 11 Author yark (2760) Entry type Definition Classification msc 20F99 Classification msc 20E99 Synonym abelian-by-finite group Synonym virtually-abelian group Related topic VirtuallyCyclicGroup Defines virtually abelian Defines abelian-by-finite Defines virtually nilpotent Defines virtually solvable Defines virtually polycyclic Defines virtually free Defines nilpotent-by-finite Defines polycyclic-by-finite Defines virtually nilpotent group Defines virtually solvable group Defines virtually polycyclic group Defines virtually free