Group whose center is a direct factor: Difference between revisions
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* [[Weaker than::Abelian group]] | * [[Weaker than::Abelian group]] | ||
* [[Weaker than::Centerless group]] | * [[Weaker than::Centerless group]] | ||
* [[Weaker than::Simple group]] | |||
* [[Weaker than::Characteristically simple group]] | |||
* [[Weaker than::Group in which every normal subgroup is a direct factor]] | |||
===Weaker properties=== | ===Weaker properties=== | ||
Latest revision as of 21:39, 24 September 2009
This article defines a group property: a property that can be evaluated to true/false for any given group, invariant under isomorphism
View a complete list of group properties
VIEW RELATED: Group property implications | Group property non-implications |Group metaproperty satisfactions | Group metaproperty dissatisfactions | Group property satisfactions | Group property dissatisfactions
Definition
A group whose center is a direct factor is a group satisfying the following equivalent conditions:
- It is the direct product of a centerless group and an abelian group.
- Its center is a permutably complemented subgroup.
- Its center is a complemented normal subgroup.
- Its center is a direct factor.
Relation with other properties
Stronger properties
- Abelian group
- Centerless group
- Simple group
- Characteristically simple group
- Group in which every normal subgroup is a direct factor
Weaker properties
- Group whose center is an AEP-subgroup
- Group having an automorphism whose restriction to the center is the inverse map
Opposite properties
- Group whose center is normality-large
- (non-abelian) nilpotent group