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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



$ F$-stability in finite groups

Authors: U. Meierfrankenfeld and B. Stellmacher
Journal: Trans. Amer. Math. Soc. 361 (2009), 2509-2525
MSC (2000): Primary 20E25
Published electronically: December 16, 2008
MathSciNet review: 2471927
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Abstract: Let $ G$ be a finite group, $ S \in \mathit {Syl}_p(G)$, and $ \mathcal S$ be the set subgroups containing $ S$. For $ M \in \mathcal S$ and $ V = \Omega_1Z(O_p(M))$, the paper discusses the action of $ M$ on $ V$. Apart from other results, it is shown that for groups of parabolic characteristic $ p$ either $ S$ is contained in a unique maximal $ p$-local subgroup, or there exists a maximal $ p$-local subgroup in $ M \in \mathcal S$ such that $ V$ is a nearly quadratic 2F-module for $ M$.

References [Enhancements On Off] (What's this?)

  • [BHS] D. Bundy, N. Hebbinghaus, B. Stellmacher, The local $ C(G,T)$-theorem, J. Algebra 300 (2006), 741-789. MR 2228220 (2007c:20050)
  • [GLM] R. M. Guralnick, R. Lawther, G. Malle, The $ 2F$-modules for nearly simple groups, J. Algebra 307 (2007), 643-676. MR 2275366 (2007k:20098)
  • [GM1] R. M. Guralnick, G. Malle, Classification of $ 2F$-modules, I, J. Algebra 257 (2002), 348-372. MR 1947326 (2003m:20008)
  • [GM2] R. M. Guralnick, G. Malle, Classification of $ 2F$-modules, II, Finite Groups 2003, 117-183, Walter de Gruyter and Co., Berlin, 2004. MR 2125071 (2006b:20062)
  • [KS] H. Kurzweil, B. Stellmacher, The theory of finite groups, Springer Universitext, New York, 2004, xii+387 pp. MR 2014408 (2004h:20001)
  • [L] R. Lawther, Abelian sets of roots and $ 2$-ranks, J. Algebra 307 (2007), 614-642. MR 2275365 (2008b:20013)
  • [MSS] U. Meierfrankenfeld, B. Stellmacher, G. Stroth, The structure theorem, in preparation.
  • [PPS] C. Parker, G. Parmeggiani, B. Stellmacher, The $ P!$-theorem, J. Algebra 263 (2003), 17-58. MR 1974077 (2004f:20033)
  • [Ste] B. Stellmacher, On the $ 2$-local structure of finite groups, in groups, combinatorics and geometry, LMS Lecture Notes Series 165 (1992), Cambridge University Press. MR 1200259 (94d:20027)

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Additional Information

U. Meierfrankenfeld
Affiliation: Department of Mathematics, Michigan State University, East Lansing, Michigan 48840

B. Stellmacher
Affiliation: Mathematisches Seminar, Christian-Albrechts-Universität, D24098 Kiel, Germany

Received by editor(s): May 16, 2006
Received by editor(s) in revised form: May 3, 2007
Published electronically: December 16, 2008
Article copyright: © Copyright 2008 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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