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.