Join of abelian subgroups of maximum rank

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

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