IJPAM: Volume 79, No. 1 (2012)
A UNIQUE IRREDUCIBLE CHARACTER
OF A GIVEN DEGREE
Department of Mathematics
Portland, OR 97202-8199, USA
Abstract. It has been conjectured that PSL, the projective special linear group of matrices over a field of order , is the only non-solvable group satisfying the property that it has a unique irreducible complex character of degree and every other irreducible complex character is such that its degree is relatively prime to . (Such a 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 is a non-abelian simple group and that when , an odd prime, itself is a non-abelian simple group, and is such that its -sylow subgroup is a cyclic group of order and equals its centralizer and that all involutions in 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