# Difference between revisions of "Join of abelian subgroups of maximum order"

(New page: ==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...) |
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+ | {{prime-parametrized subgroup-defining function}} | ||

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==Definition== | ==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]]. | 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]]. | ||

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+ | ===As a characteristic p-functor=== | ||

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+ | 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]]. |

## Latest revision as of 20:15, 2 March 2009

Template:Prime-parametrized subgroup-defining function

## Definition

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.