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Transactions of the American Mathematical Society

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

 

 

A generlization of Feit's theorem


Author: J. H. Lindsey
Journal: Trans. Amer. Math. Soc. 155 (1971), 65-75
MSC: Primary 20.25
DOI: https://doi.org/10.1090/S0002-9947-1971-0279173-6
MathSciNet review: 0279173
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Abstract: This paper is part of a doctoral thesis at Harvard University. The title of the thesis is Finite linear groups in six variables.

Using the methods of this paper, I believe that I can prove that if $ p$ is a prime greater than five with $ p \equiv - 1\pmod 4$, and $ G$ is a finite group with faithful complex representation of degree smaller than $ 4p/3$ for $ p > 7$ and degree smaller than 9 for $ p = 7$, then $ G$ has a normal $ p$-subgroup of index in $ G$ divisible at most by $ {p^2}$. These methods are particularly effective when there is nontrivial intersection of $ p$-Sylow subgroups. In fact, if the current work people are doing on the trivial intersection case can be extended, it should be possible to show that, for $ p$ a prime and $ G$ a finite group with a faithful complex representation of degree less than $ 3(p - 1)/2,G$ has a normal $ p$-subgroup of index in $ G$ divisible at most by $ {p^2}$. (It may be possible to show that the index is divisible at most by $ p$ if the representation is primitive and has degree unequal to $ p$.)


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DOI: https://doi.org/10.1090/S0002-9947-1971-0279173-6
Keywords: Quasiprimitive, homogeneous space, $ PSL(2,p)$
Article copyright: © Copyright 1971 American Mathematical Society