Finite groups with exactly one composite character degree (Q2816895)

This paper is another contribution to the study of how restrictions on complex character degrees influence the group structure. In the 1960s, \textit{I. M. Isaacs} and \textit{D. S. Passman} characterized all finite groups whose nonlinear character degrees were all prime numbers [Pac. J. Math. 24, 467--510 (1968; Zbl 0155.05502); ibid. 15, 877--903 (1965; Zbl 0132.01902)].NEWLINENEWLINEIn the paper under review, the authors study finite groups \(G\) with exactly one nonlinear character degree that is not a prime. They prove that then one of the following is true: (1) \(G\) is solvable with \(|\mathrm{cd}(G)|\leq 4\); (2) \(G\) is solvable with \(\mathrm{cd}(G)=\{1,f,p,q,qf\}\) for distinct primes \(f\), \(p\), \(q\); or (3) up to an abelian direct factor, \(G\) is isomorphic to the alternating group \(A_5\). They remark that the proof depends on CFSG.