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Some criteria for the nonexistence of certain finite linear groups


Author: Harvey I. Blau
Journal: Proc. Amer. Math. Soc. 43 (1974), 283-286
MSC: Primary 20C20
DOI: https://doi.org/10.1090/S0002-9939-1974-0332946-1
MathSciNet review: 0332946
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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 $ \vert N:C\vert = p - 1$, then $ d \geqq p - 1$.


References [Enhancements On Off] (What's this?)

  • [1] H. I. Blau, Under the degree of some finite linear groups, Trans. Amer. Math. Soc. 155 (1971), 95-113. MR 43 #367. MR 0274604 (43:367)
  • [2] W. Feit, Groups with a cyclic Sylow subgroup, Nagoya Math. J. 27 (1966), 571-584. MR 33 #7404. MR 0199255 (33:7404)
  • [3] J. H. Lindsey II, On a six dimensional projective representation of the Hall-Janko group, Pacific J. Math. 35 (1970), 175-186. MR 42 #7769. MR 0272888 (42:7769)

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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1974-0332946-1
Keywords: Indecomposable modular representation, small degree, cyclic Sylow $ p$-subgroup
Article copyright: © Copyright 1974 American Mathematical Society

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