Subgroup with abelianization of maximum order

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This article is about a maximality notion among subgroups, related to abelianness or small class, in a group of prime power order.
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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 $P$.

Relation with other properties

Stronger properties

• Minimal subgroup with abelianization of maximum order