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Join of abelian subgroups of maximum order

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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.