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 Mat. Zametki: Year: Volume: Issue: Page: Find

 Mat. Zametki, 2016, Volume 99, Issue 1, Pages 121–139 (Mi mz10184)

On Automorphisms of Irreducible Linear Groups with an Abelian Sylow $2$-Subgroup

Institute of Mathematics of the National Academy of Sciences of Belarus

Abstract: Let $\Gamma=AG$ be a finite group, where $G\triangleleft\Gamma$, $(|G|,|A|)=1$, and let $A$ be a nonprimary subgroup of odd order which is not normal in $\Gamma$. The Sylow $2$-subgroup of the group $G$ is Abelian, and $C_G(a)=C_G(A)$ for every element $a\in A^{#}$, where $A^{#}$ stands for the set of nonidentity elements of $A$. Suppose that the group $G$ has a faithful irreducible complex character of degree $n$ which is $a$-invariant for at least one element $a\in A^{#}$. In the present paper, it is proved that $n$ is divisible by a power of a prime with exponent $f>1$ such that $f\equiv -1$ or $1 (\operatorname{mod}|A|)$.

Keywords: irreducible linear group, Abelian Sylow $2$-subgroup, faithful, irreducible complex character.

DOI: https://doi.org/10.4213/mzm10184

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English version:
Mathematical Notes, 2016, 99:1, 138–154

Bibliographic databases:

UDC: 512.542
Revised: 17.06.2015

Citation: A. A. Yadchenko, “On Automorphisms of Irreducible Linear Groups with an Abelian Sylow $2$-Subgroup”, Mat. Zametki, 99:1 (2016), 121–139; Math. Notes, 99:1 (2016), 138–154

Citation in format AMSBIB
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