Some criteria for the nonexistence of certain finite linear groups

Abstract:

Let $p$ be a prime and $G$ a finite group, not of type ${L_2}(p)$, with a cyclic Sylow $p$-subgroup $P$. Assume that $G = G’$. The purpose of this note is to put some rather stringent lower bounds on the degree $d$ of a faithful indecomposable representation of $G$ over a field of characteristic $p$ given certain conditions on the normalizer $N$ and the centralizer $C$ of $P$ in $G$. In particular, if the center of $G$ has order 2 and $|N:C| = p - 1$, then $d \geqq p - 1$.