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.