Some Nonsolvable Character Degree Sets

Abstract

For positive integer k and nonabelian simple group S, let $S^{k}$ be the direct product of k copies of S. We conjecture that all finite groups G with $\mathrm{cd}(G)=\mathrm{cd}(S^{k})$ are quasi perfect groups (that is; $G'=G''$) and hence nonsolvable groups, where $\mathrm{cd}(G)$ is the set of irreducible character degrees of G. In this paper, we prove this conjecture for $S\in \{\mathrm{PSL}_{2}(p^{f}), \mathrm{PSL}_{2}(2^{f}), \mathrm{Sz}(q)\}$, where $p>2$ is an odd prime number such that $p^{f}>5$ and $p^{f}\pm 1\not \mid 2^{k}$, and $q=2^{2n+1}\geqslant 8$.