Join of abelian subgroups of maximum order: Difference between revisions

Latest revision as of 20:15, 2 March 2009

Template:Prime-parametrized subgroup-defining function

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 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.

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 ==Definition==
-Let <math>P</math> be a [[group of prime power order]]. The '''join of Abelian subgroups of maximum order''' in <math>P</math>, sometimes denoted <math>J(P)</math> and also termed the '''Thompson subgroup''', is defined as the subgroup of <math>P</math> generated by all [[defining ingredient::abelian subgroup of maximum order|abelian subgroups of maximum order]] in <math>P</math>.
+Let <math>P</math> be a [[group of prime power order]]. The '''join of Abelian subgroups of maximum order''' in <math>P</math>, sometimes denoted <math>J(P)</math> and also termed the '''Thompson subgroup''' or the '''Thompson J-subgroup''', is defined as the subgroup of <math>P</math> generated by all [[defining ingredient::abelian subgroup of maximum order|abelian subgroups of maximum order]] in <math>P</math>.
 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 <math>p</math>-group <math>P</math>, the subgroup <math>J(P)</math> is also nontrivial, since <math>P</math> has nontrivial abelian subgroups. Thus, this is a [[characteristic p-functor]], and in particular, is a [[conjugacy functor]].  A closely related, and extremely important, <math>p</math>-functor is the [[ZJ-functor]] whose many properties were explored by [[Glauberman]].