Finite groups whose real-valued irreducible characters have prime power degrees
Group characters always encode various important information of the structure of a finite group. On the one hand, finite groups whose irreducible character degrees are all prime powers were classified by \emph{O. Manz} in [J. Algebra 94, 211--255 (1985; Zbl 0596.20007); J. Algebra 96, 114--119 (1985; Zbl 0567.20004)]. On the other hand, \emph{S. Dolfi} et al. [J. Algebra 320, No. 5, 2181--2195 (2008; Zbl 1176.20007)] considered finite groups whose real-valued irreducible character degrees can only be \(1\) and prime numbers. The paper extends the investigations to finite groups in which the degree of every real-valued irreducible character is a prime power. It is proved in this paper that the only possible non-abelian composition factors of such groups are \(L_2(4)\) and \(L_2(8)\). The paper also shows that, if a solvable group satisfies the condition, then the real-valued character degrees cannot be powers of more than two distinct primes. Finally, the paper gives a criterion for a finite group to be solvable in terms of nonlinear rational irreducible characters, generalizing and strengthening one of the main results in the paper of Dolfi et al. [loc. cit.].
scientific article; zbMATH DE number 7474385
scientific article