Group having an abelian normal non-central subgroup
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 having an Abelian normal non-central subgroup is a group satisfying the following equivalent conditions:
- There exists an Abelian normal subgroup not contained inside the center
- There exists an Abelian normal subgroup properly containing the center
Relation with other properties
Stronger properties
- non-Abelian nilpotent group
- non-Abelian supersolvable group
- non-Abelian group in which maximal among Abelian normal implies self-centralizing