On linear groups over finite fields

Zhang, Ji Ping

doi:10.1090/S0002-9939-1990-1028297-1

On linear groups over finite fields
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Proc. Amer. Math. Soc. 110 (1990), 53-57 Request permission

Abstract:

Let $G$ be a finite group with an Abelian Sylow $p$-subgroup $P$ $(p > 5)$, and $F$, a finite field of characteristic $p$. Set $H = {O^{p’}}(G)$. If $G$ has a faithful FG-module $M$ such that ${\dim _F}M < p - 2$, then one of the following is true: (a) $P$ is normal in $G$, (b) $H/Z(H) \approx { \oplus _{i \leq t}}{L_2}({p^{{n_i}}})$, where ${n_i}$ and $t$ are positive integers and $2t < p - 2$, (c) $p = 7{\text {or}}11$ and $H \approx 2.{A_7}$ or ${J_1}$, respectively, ${\dim _F}M \geq p - 4$.

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