Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

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

 

 

$ p$-solvable linear groups of finite order


Author: David L. Winter
Journal: Trans. Amer. Math. Soc. 157 (1971), 155-160
MSC: Primary 20.40
MathSciNet review: 0276345
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: The purpose of this paper is to prove the following result.

Theorem. Let $ p$ be an odd prime and let $ G$ be a finite $ p$-solvable group. Assume that $ G$ has a faithful representation of degree $ n$ over a field of characteristic zero or over a perfect field of characteristic $ p$. Let $ P$ be a Sylow $ p$-subgroup of $ G$ and let $ {O_p}(G)$ be the maximal normal $ p$-subgroup of $ G$. Then $ \vert P:{O_p}(G)\vert \leqq {p^{{\lambda _p}(n)}}$ where

\begin{displaymath}\begin{array}{*{20}{c}} {{\lambda _p}(n) = \sum\limits_{i = 0... ...ight]} \quad if\;p\;is\;not\;a\;Fermat\;prime.} \\ \end{array} \end{displaymath}


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

  • [1] John D. Dixon, Normal 𝑝-subgroups of solvable linear groups, J. Austral. Math. Soc. 7 (1967), 545–551. MR 0230815
  • [2] George Glauberman, Correspondences of characters for relatively prime operator groups., Canad. J. Math. 20 (1968), 1465–1488. MR 0232866
  • [3] P. Hall and Graham Higman, On the 𝑝-length of 𝑝-soluble groups and reduction theorems for Burnside’s problem, Proc. London Math. Soc. (3) 6 (1956), 1–42. MR 0072872
  • [4] L. G. Kovács, On finite soluble groups, Math. Z. 103 (1968), 37–39. MR 0223458
  • [5] I. Schur, Über eine Klasse von endlichen Gruppen linearer Substitutionen, S.-B. Preuss. Akad. Wiss. 1905, 77-91.
  • [6] Richard G. Swan, The Grothendieck ring of a finite group, Topology 2 (1963), 85–110. MR 0153722

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 20.40

Retrieve articles in all journals with MSC: 20.40


Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1971-0276345-1
Keywords: $ p$-solvable linear group, normal $ p$-subgroup
Article copyright: © Copyright 1971 American Mathematical Society