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virtually abelian group (Definition)

A group $G$ is virtually abelian (or abelian-by-finite) if it has an abelian subgroup of finite index.

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).



"virtually abelian group" is owned by yark. [ full author list (2) | owner history (1) ]
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See Also: virtually cyclic group

Other names:  abelian-by-finite group, virtually-abelian group
Also defines:  virtually abelian, abelian-by-finite, virtually nilpotent, virtually solvable, virtually polycyclic, virtually free, nilpotent-by-finite, polycyclic-by-finite, virtually nilpotent group, virtually solvable group, virtually polycyclic group, virtually free group, nilpotent-by-finite group, polycyclic-by-finite group, virtually-free
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Cross-references: finite groups, infinite, converse, normal subgroup, finite, abelian, group
There are 4 references to this entry.

This is version 8 of virtually abelian group, born on 2004-09-13, modified 2006-08-17.
Object id is 6168, canonical name is VirtuallyAbelian.
Accessed 12905 times total.

Classification:
AMS MSC: 20E99 (Group theory and generalizations :: Structure and classification of infinite or finite groups :: Miscellaneous)
 20F99 (Group theory and generalizations :: Special aspects of infinite or finite groups :: Miscellaneous)
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