IJPAM: Volume 79, No. 1 (2012)

FINITE NON-SOLVABLE GROUPS HAVING
A UNIQUE IRREDUCIBLE CHARACTER
OF A GIVEN DEGREE

Venkata Rao Potluri
Department of Mathematics
Reed College
Portland, OR 97202-8199, USA


Abstract. It has been conjectured that PSL$(2,q)$, the projective special linear group of $2\times 2$ matrices over a field of order $q$, is the only non-solvable group satisfying the property that it has a unique irreducible complex character $\chi$ of degree $m>1$ and every other irreducible complex character is such that its degree is relatively prime to $m$. (Such a $\chi$ is a particular case of the Steinberg character of finite Chevalley groups.) In this paper, we consider finite non-solvable groups satisfying the above property and show that the derived group $G'$ is a non-abelian simple group and that when $\chi (1) = p$, $p$ an odd prime, $G$ itself is a non-abelian simple group, and is such that its $p$-sylow subgroup $P$ is a cyclic group of order $p$ and equals its centralizer and that all involutions in $G$ are conjugate.

Received: April 1, 2012

AMS Subject Classification: 20C15

Key Words and Phrases: finite non-solvable groups, Chevalley groups, Steinberg character, irreducible complex characters

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Source: International Journal of Pure and Applied Mathematics
ISSN printed version: 1311-8080
ISSN on-line version: 1314-3395
Year: 2012
Volume: 79
Issue: 1