# 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** or the **Thompson J-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.

### As a characteristic p-functor

For a nontrivial -group , the subgroup is also nontrivial, since has nontrivial abelian subgroups. Thus, this is a characteristic p-functor, and in particular, is a conjugacy functor. A closely related, and extremely important, -functor is the ZJ-functor whose many properties were explored by Glauberman.