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单词 VirtuallyAbelianGroup
释义

virtually abelian group


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

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).

Titlevirtually abelian group
Canonical nameVirtuallyAbelianGroup
Date of creation2013-03-22 14:35:58
Last modified on2013-03-22 14:35:58
Owneryark (2760)
Last modified byyark (2760)
Numerical id11
Authoryark (2760)
Entry typeDefinition
Classificationmsc 20F99
Classificationmsc 20E99
Synonymabelian-by-finite group
Synonymvirtually-abelian group
Related topicVirtuallyCyclicGroup
Definesvirtually abelian
Definesabelian-by-finite
Definesvirtually nilpotent
Definesvirtually solvable
Definesvirtually polycyclic
Definesvirtually free
Definesnilpotent-by-finite
Definespolycyclic-by-finite
Definesvirtually nilpotent group
Definesvirtually solvable group
Definesvirtually polycyclic group
Definesvirtually free
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更新时间:2025/5/5 0:01:04