# Join of elementary abelian subgroups of maximum order

From Groupprops

This article defines a subgroup-defining function, viz., a rule that takes a group and outputs a unique subgroup

View a complete list of subgroup-defining functions OR View a complete list of quotient-defining functions

## Definition

Suppose is a prime number and is a finite p-group. The **join of elementary abelian subgroups of maximum order** in , denoted , is defined as the join of all the elementary abelian subgroups that have maximum order among elementary abelian subgroups of .

This is one of the three Thompson subgroups. In some contexts, it is referred to as *the* Thompson subgroup.