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

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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UDC: 512.547.214

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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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
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