# Abelian subgroup of maximum rank

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

## Contents

## Definition

Suppose is a group of prime power order. A subgroup of is termed an **abelian subgroup of maximum rank** if is an abelian subgroup of and the rank of (i.e., the minimum number of elements needed to generate ) is the maximum among the ranks of all abelian subgroups of .

Note that abelian subgroups of maximum rank need not be maximal among abelian subgroups.

The join of all such subgroups is termed the join of abelian subgroups of maximum rank, and is one of the three Thompson subgroups often considered for groups of prime power order.

## Relation with other properties

### Stronger properties

- Maximal among abelian subgroups of maximum rank: These are the maximal elements with respect to inclusion among abelian subgroups of maximum rank.
- Elementary abelian subgroup of maximum order