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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

On linear groups over finite fields


Author: Ji Ping Zhang
Journal: Proc. Amer. Math. Soc. 110 (1990), 53-57
MSC: Primary 20C20
DOI: https://doi.org/10.1090/S0002-9939-1990-1028297-1
MathSciNet review: 1028297
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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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DOI: https://doi.org/10.1090/S0002-9939-1990-1028297-1
Article copyright: © Copyright 1990 American Mathematical Society

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