$p$-solvable linear groups of finite order
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- by David L. Winter PDF
- Trans. Amer. Math. Soc. 157 (1971), 155-160 Request permission
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 $|P:{O_p}(G)| \leqq {p^{{\lambda _p}(n)}}$ where \[ \begin {array}{*{20}{c}} {{\lambda _p}(n) = \sum \limits _{i = 0}^\infty {\left [ {\frac {n}{{(p - 1){p^i}}}} \right ]} \quad if\;p\;is\;a\;Fermat\;prime,} \\ { = \sum \limits _{i = 1}^\infty {\left [ {\frac {n}{{{p^i}}}} \right ]} \quad if\;p\;is\;not\;a\;Fermat\;prime.} \\ \end {array} \]References
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Additional Information
- © Copyright 1971 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 157 (1971), 155-160
- MSC: Primary 20.40
- DOI: https://doi.org/10.1090/S0002-9947-1971-0276345-1
- MathSciNet review: 0276345