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Transactions of the American Mathematical Society
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
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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}


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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9947-1971-0276345-1
PII: S 0002-9947(1971)0276345-1
Keywords: $ p$-solvable linear group, normal $ p$-subgroup
Article copyright: © Copyright 1971 American Mathematical Society