Abelian hereditarily normal subgroup

Definition
A subgroup $$H$$ of a group $$G$$ is termed an abelian hereditarily normal subgroup or abelian transitively normal subgroup of $$G$$ if it satisfies the following equivalent conditions:


 * 1) $$H$$ is abelian as a group and is a hereditarily normal subgroup of $$G$$.
 * 2) $$H$$ is abelian as a group and is a transitively normal subgroup of $$G$$.
 * 3) $$H$$ is an defining ingredient::abelian normal subgroup of $$G$$ and the induced action of the quotient group $$G/H$$ on $$H$$ is by power automorphisms.