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Model. Anal. Inform. Sist., 2015, Volume 22, Number 4, Pages 483–499 (Mi mais454)  

This article is cited in 1 scientific paper (total in 1 paper)

On finite groups with an irreducible character large degree

L. S. Kazarin, S. S. Poiseeva

P.G. Demidov Yaroslavl State University, Sovetskaya str., 14, Yaroslavl, 150000, Russia

Abstract: Let $G$ be a finite nontrivial group with an irreducible complex character $\chi$ of degree $d=\chi(1)$. It is known from the orthogonality relation that the sum of the squares of degrees of irreducible characters of $G$ is equal to the order of $G$. N. Snyder proved that if $|G|=d(d+e)$, then the order of $G$ is bounded in terms of $e$, provided $e>1$. Y. Berkovich proved that in the case $e=1$ the group $G$ is Frobenius with the complement of order $d$. We study a finite nontrivial group $G$ with an irreducible complex character $\Theta$ such that $|G|\leq2\Theta(1)^2$ and $\Theta(1)=pq$, where $p$ and $q$ are different primes. In this case we prove that $G$ is solvable groups with abelian normal subgroup $K$ of index $pq$. We use the classification of finite simple groups and prove that the simple nonabelian group whose order is divisible by a prime $p$ and of order less than $2p^4 $ is isomorphic to $L_2(q), L_3(q), U_3(q), Sz(8), A_7, M_{11}$ or $J_1$.

Keywords: finite group, character of a finite group, irreducible character degree of a finite group.

Funding Agency Grant Number
Russian Foundation for Basic Research 13-01-00469


DOI: https://doi.org/10.18255/1818-1015-2015-4-483-499

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Bibliographic databases:

UDC: 512.547.214
Received: 06.07.2015

Citation: L. S. Kazarin, S. S. Poiseeva, “On finite groups with an irreducible character large degree”, Model. Anal. Inform. Sist., 22:4 (2015), 483–499

Citation in format AMSBIB
\Bibitem{KazPoi15}
\by L.~S.~Kazarin, S.~S.~Poiseeva
\paper On finite groups with an irreducible character large degree
\jour Model. Anal. Inform. Sist.
\yr 2015
\vol 22
\issue 4
\pages 483--499
\mathnet{http://mi.mathnet.ru/mais454}
\crossref{https://doi.org/10.18255/1818-1015-2015-4-483-499}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=3418468}
\elib{http://elibrary.ru/item.asp?id=24273049}


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    Citing articles on Google Scholar: Russian citations, English citations
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    This publication is cited in the following articles:
    1. S. S. Poiseeva, “O stroenii konechnykh grupp s bolshim neprivodimym kharakterom stepeni $p^2q$”, Matematicheskie zametki SVFU, 23:3 (2016), 81–90  mathnet  elib
  • Моделирование и анализ информационных систем
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