Join of abelian subgroups of maximum order

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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, 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.