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