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 $$q=2^{2n+1}\geqslant 8$$.