A generlization of Feit’s theorem

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 $p$.)