Finite groups whose irreducible characters vanish on prime power order. (Q2908972)

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 Generalizing Theorem B of \textit{D. Bubboloni, S. Dolfi} and \textit{P. Spiga}, [J. Pure Appl. Algebra 213, No. 3, 370-376 (2009; Zbl 1162.20004)], the authors prove that if every nonlinear character \(\chi\) of \(G\) vanishes only on elements of prime power order and if in addition every nonlinear irreducible character \(\chi\) vanishes on some \(p\)-elements for a fixed prime \(p\), then either \(G\) is a \(p\)-group, or \(Z(G)=O_p(G)\), and \(G/Z(G)\) is a Frobenius group with complement \(P/O_p(G)\) for a Sylow \(p\)-subgroup \(P\) of \(G\). They also prove that if the orders of elements of \(G\) on which some irreducible character vanishes are just three different primes, then \(G\) is isomorphic to \(A_5\).