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virtually abelian group
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(Definition)
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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 <</SPAN>#57#>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).
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"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 5 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 16003 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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Pending Errata and Addenda
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