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
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
|Property||Meaning||Proof of implication||Proof of strictness (reverse implication failure)||Intermediate notions|
|abelian group||nilpotency class at most one||(obvious)||2-Engel group|FULL LIST, MORE INFO|
|group of nilpotency class two||nilpotency class at most two||class two implies generated by abelian normal subgroups||finite group generated by abelian normal subgroups may have arbitrarily large nilpotency class||2-Engel group|FULL LIST, MORE INFO|
|Levi group||every element is in an abelian normal subgroup, equivalently, a 2-Engel group||Levi implies generated by abelian normal subgroups|||FULL LIST, MORE INFO|
- 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.