Join of abelian subgroups of maximum order

Definition

Let $P$ be a group of prime power order. The join of Abelian subgroups of maximum order in $P$, sometimes denoted $J(P)$ and also termed the Thompson subgroup or the Thompson J-subgroup, is defined as the subgroup of $P$ generated by all abelian subgroups of maximum order in $P$.

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