Finite groups whose irreducible characters vanish only on elements of prime power order. (Q2920202)

By a result of \textit{G. Malle, G. Navarro} and \textit{J. B. Olsson} [J. Group Theory 3, No. 4, 353-368 (2000; Zbl 0965.20003)], every nonlinear irreducible character \(\chi\) of a finite group \(G\) vanishes on some element of prime power order of \(G\). In the paper under review, the authors are interested in the following question: Assume that every irreducible character vanishes only on elements of prime power order, what can be said about the structure of \(G\)?NEWLINENEWLINE The authors prove several results on this question. For instance, if the solvable radical of \(G\) is trivial, then \(G\) is isomorphic to one of the following groups: \(L_2(q)\) for \(q=5,7,8,9,17\), \(L_3(4)\), \(Sz(8)\), \(Sz(32)\), or a specific one of the three subgroups of index 2 in \(\Aut(L_2(9))\). The proof depends on CFSG.NEWLINENEWLINE The authors also describe solvable groups whose irreducible characters vanish only on elements of prime order. Moreover, they prove that \(G\) is a finite group such that the set of orders of elements on which some irreducible character vanishes is \(\{2,3,5\}\), then \(G\cong A_5\).