# Join of abelian subgroups of maximum order

## Definition

Let be a group of prime power order. The **join of Abelian subgroups of maximum order** in , sometimes denoted and also termed the **Thompson subgroup**, is defined as the subgroup of generated by all abelian subgroups of maximum order in .

Note that the term *Thompson subgroup* is also used for the join of abelian subgroups of maximum rank and for the join of elementary abelian subgroups of maximum order.