# Subgroup with abelianization of maximum order

From Groupprops

This article defines a subgroup property: a property that can be evaluated to true/false given a group and a subgroup thereof, invariant under subgroup equivalence. View a complete list of subgroup properties[SHOW MORE]

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

View other such notions

## Definition

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