Subgroup with abelianization of maximum order

Definition

Let $P$ be a group of prime power order. A subgroup $B$ of $P$ is termed a subgroup with abelianization of maximum order if the order of the abelianization of $B$ (i.e., the quotient of $B$ by its commutator subgroup) is greater than or equal to the order of the abelianization of any subgroup of $P$.