NSCFN-realizable group: Difference between revisions
(New page: {{wikilocal}} {{group property}} ==Definition== A group is termed '''NSCFN-realizable''' if it can be embedded as a NSCFN-subgroup of some group. ==Relation with other propertie...) |
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==Definition== | ==Definition== | ||
A [[group]] is termed '''NSCFN-realizable''' if it can be embedded as a [[NSCFN-subgroup]] of some group. | A [[group]] <math>G</math> is termed '''NSCFN-realizable''' if it satisfies the following equivalent conditions: | ||
* <math>G</math> can be embedded as a [[NSCFN-subgroup]] of some group. | |||
* Given any homomorphism <math>\alpha: L \to \operatorname{Out}(G)</math>, there exists a group <math>K</math> containing <math>G</math> as a normal subgroup with <math>K/G</math> isomorphic to <math>L</math>, and the [[quotient group maps to outer automorphism group of normal subgroup|induced outer action]] of <math>G</math> on <math>L</math> is <math>\alpha</math>. | |||
* The [[outer action cohomology class]] in <math>H^3(\operatorname{Out}(G),Z(G))</math> is trivial. | |||
==Relation with other properties== | ==Relation with other properties== | ||
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* [[Weaker than::Centerless group]] | * [[Weaker than::Centerless group]] | ||
* [[Weaker than::Group in which every automorphism is inner]] | * [[Weaker than::Group in which every automorphism is inner]] | ||
==Facts== | |||
* [[Special linear group:SL(2,9) is not NSCFN-realizable]] | |||
Latest revision as of 20:22, 15 April 2009
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This article defines a group property: a property that can be evaluated to true/false for any given group, invariant under isomorphism
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Definition
A group is termed NSCFN-realizable if it satisfies the following equivalent conditions:
- can be embedded as a NSCFN-subgroup of some group.
- Given any homomorphism , there exists a group containing as a normal subgroup with isomorphic to , and the induced outer action of on is .
- The outer action cohomology class in is trivial.
Relation with other properties
Stronger properties
- Group whose center is a direct factor
- Abelian group
- Centerless group
- Group in which every automorphism is inner