Join of abelian subgroups of maximum order
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.