Abelian subnormal subgroup

Definition
A subgroup $$H$$ of a group $$G$$ is termed an abelian subnormal subgroup if $$H$$ is an abelian group as a group in its own right (or equivalently, an abelian subgroup of $$G$$) and $$H$$ is a subnormal subgroup of $$G$$.

Weaker properties
{| class="sortable" border="1" ! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions
 * Stronger than::nilpotent subnormal subgroup || || || ||
 * Stronger than::nilpotent subnormal subgroup || || || ||