Thompson transitivity theorem: Difference between revisions

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In mathematical [[finite group theory]], the '''Thompson transitivity theorem''' gives conditions under which the [[Centralizer and normalizer|centralizer]] of an [[abelian subgroup]] ''A'' acts [[Group_action#Transitivity_properties|transitively]] on certain subgroups normalized by ''A''. It originated in the proof of the [[Feit–Thompson theorem|odd order theorem]] by {{harvs|txt|last=Feit|last2=Thompson|author2-link=John G. Thompson|year=1963}}, where it was used to prove the [[Thompson uniqueness theorem]].

==Statement==

Suppose that ''G'' is a finite group and ''p'' a [[Prime number|prime]] such that all [[P-local subgroup|''p''-local subgroups]] are [[p-constrained|''p''-constrained]]. If ''A'' is a self-centralizing normal abelian subgroup of a ''p''-Sylow subgroup such that ''A'' has rank at least 3, then the centralizer C_''G''(''A'') act transitively on the maximal ''A''-invariant ''q'' subgroups of ''G'' for any prime ''q'' ≠ ''p''.

Suppose that ''G'' is a finite group and ''p'' a prime such that all ''p''-local subgroups are [[p-constrained]]. If ''A'' is a self-centralizing normal abelian subgroup of a ''p''-Sylow subgroup such that ''A'' has rank at least 3, then the centralizer C_''G''(''A'') act transitively on the maximal ''A''-invariant ''q'' subgroups of ''G'' for any prime ''q'' ≠ ''p''.

==References==

*{{Citation | last1=Bender | first1=Helmut | last2=Glauberman | first2=George | author2-link=George Glauberman | title=Local analysis for the odd order theorem | publisher=[[Cambridge University Press]] | series=London Mathematical Society Lecture Note Series | isbn=978-0-521-45716-3 | mr=1311244 | year=1994 | volume=188}}

*{{Citation | last1=Feit | first1=Walter | author1-link=Walter Feit | last2=Thompson | first2=John G. | author2-link=John G. Thompson | title=Solvability of groups of odd order | url=http://projecteuclid.org/Dienst/UI/1.0/Journal?authority=euclid.pjm&issue=1103053941 | mr=0166261 | year=1963 | journal=Pacific Journal of Mathematics | issn=0030-8730 | volume=13 | pages=775–1029| doi=10.2140/pjm.1963.13.775 | doi-access=free }}

*{{Citation | last1=Gorenstein | first1=D. | author1-link=Daniel Gorenstein | title=Finite groups | url=http://www.ams.org/bookstore-getitem/item=CHEL-301-H | publisher=Chelsea Publishing Co. | ___location=New York | edition=2nd | isbn=978-0-8284-0301-6 | idmr={{MathSciNet | id = 569209}} | year=1980}}

[[Category:Theorems inabout groupfinite theorygroups]]

*{{Citation | last1=Bender | first1=Helmut | last2=Glauberman | first2=George | author2-link=George Glauberman | title=Local analysis for the odd order theorem | publisher=[[Cambridge University Press]] | series=London Mathematical Society Lecture Note Series | isbn=978-0-521-45716-3 | mr=1311244 | year=1994 | volume=188}}

*{{Citation | last1=Feit | first1=Walter | author1-link=Walter Feit | last2=Thompson | first2=John G. | author2-link=John G. Thompson | title=Solvability of groups of odd order | url=http://projecteuclid.org/Dienst/UI/1.0/Journal?authority=euclid.pjm&issue=1103053941 | mr=0166261 | year=1963 | journal=Pacific Journal of Mathematics | issn=0030-8730 | volume=13 | pages=775–1029}}

*{{Citation | last1=Gorenstein | first1=D. | author1-link=Daniel Gorenstein | title=Finite groups | url=http://www.ams.org/bookstore-getitem/item=CHEL-301-H | publisher=Chelsea Publishing Co. | ___location=New York | edition=2nd | isbn=978-0-8284-0301-6 | id={{MathSciNet | id = 569209}} | year=1980}}

{{abstract-algebra-stub}}

[[Category:Finite groups]]

[[Category:Theorems in group theory]]