virtually abelian group

A group G is virtually abelian (or abelian-by-finite) if it has an abelianMathworldPlanetmath subgroupMathworldPlanetmathPlanetmath ( of finite index (

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

These notions are obviously only of relevance to infinite groups, as all finite groupsMathworldPlanetmath 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