Behrooz Khosravi, Behnam Khosravi, Bahman Khosravi : A new characterization for some extensions of $\text{PSL}(2,p)$ by order and some character degrees, 267-275

Abstract:

Many authors were recently concerned with the following question: What can be said about the structure of a finite group $G$, if some information is known about the arithmetical structure of the degrees of the irreducible characters of $G$?

Let $G$ be a finite group and $X_1(G)$ be the set of all irreducible complex character degrees of $G$ counting multiplicities.

Let $p$ be an odd prime number and M = PGL(2,p), M = Z₂xPSL(2,p) or $M={\rm SL}(2,p)$. In this paper we prove that $M$ is uniquely determined by its order and some information on its character degrees. As a consequence of our results we prove that if $G$ is a finite group such that $X_1(G)=X_1(M)$, then $G\cong M$. This implies that $M$ is uniquely determined by the structure of its complex group algebra.

Key Words: Character degrees, order, projective special linear group, characterization

2010 Mathematics Subject Classification: Primary 20C15
Secondary 20D05, 20D60