Abelian characteristic subgroup

Definition
A subgroup $$H$$ of a group $$G$$ is termed an abelian characteristic subgroup if $$H$$ is abelian as a group (i.e., $$H$$ is an abelian subgroup of $$G$$ and also, $$H$$ is a characteristic subgroup of $$G$$, i.e., $$H$$ is invariant under all automorphisms of $$G$$.

Facts

 * Second half of lower central series of nilpotent group comprises abelian groups: In particular, this means that for a group $$G$$ of nilpotency class $$c$$, all the subgroups $$\gamma_k(G), k \ge (c + 1)/2$$ are abelian characteristic subgroups.
 * The center of any group is an abelian characteristic subgroup.

Related group properties

 * Group in which every abelian characteristic subgroup is central