# Abelian subgroup of maximum order which is normal

Suppose $G$ is a group of prime power order, i.e., $G$ is a finite p-group for some prime number $p$. Suppose $H$ is a subgroup of $G$. We say that $H$ is an abelian subgroup of maximum order which is normal if $H$ is an abelian subgroup of maximum order in $G$ and $H$ is also a normal subgroup of $G$.