Group generated by abelian normal subgroups
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
Symbol-free definition
A group is said to be generated by Abelian normal subgroups if there exists a collection of Abelian normal subgroups which together generate the group.
Examples
The dihedral group of size eight and the quaternion group are examples of non-Abelian groups generated by Abelian normal subgroups. While the former is generated by a cyclic normal subgroup of order 4 and a Klein four-group, the latter is generated by two cyclic normal subgroups.
Relation with other properties
Stronger properties
Weaker properties
- Nilpotent group (for finite groups): For proof of the implication, refer Finite and generated by abelian normal subgroups implies nilpotent and for proof of its strictness (i.e. the reverse implication being false) refer Nilpotent not implies generated by abelian normal subgroups.