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