Finite nonsolvable groups in which only two nonlinear irreducible characters have equal degrees (Q1922895)

In [Proc. Am. Math. Soc. 115, No. 4, 955-959 (1992; Zbl 0822.20004)], \textit{Ya. Berkovich, D. Chillag} and \textit{M. Herzog} have classified finite groups in which the degrees of the nonlinear irreducible characters are distinct. A finite group in which only two nonlinear irreducible characters have equal degrees is said to be a \(D_1\)-group. The first author [J. Algebra 184, No. 2, 584-603 (1996; see the preceding review Zbl 0861.20008)] has described soluble \(D_1\)-groups. In this paper, the authors show (using the classification of finite simple groups) that \(L_2(5)\) and \(L_2(7)\) are the only nonsolvable \(D_1\)-groups. The results of \textit{W. Feit} and \textit{G. Seitz} [Ill. J. Math. 33, No. 1, 103-131 (1989; Zbl 0701.20005)] play a key role in the proof.