On a question of Brauer (Q1077524)

Let \(\chi\) be an irreducible complex character of the finite group G, and denote by f(\(\chi)\) the smallest positive integer such that the field of f(\(\chi)\)-th roots of 1 contains all values of \(\chi\). W. Feit has asked whether G must contain an element of order f(\(\chi)\). Various special cases are known. In the paper under review the authors consider the following weaker problem: If p,q,r are different primes and if a,b,c are nonnegative integers such that \(m=p^ aq^ br^ c\) divides f(\(\chi)\), does G contain an element of order m? It follows from results of R. Brauer that in a counterexample \(a=1\) if p is odd, and \(a\geq 2\) if \(p=2\). The authors prove that, for \(\chi\) (1) odd, a minimal counterexample must be a perfect central extension of a simple group S. Moreover, they rule out the cases where S is sporadic, alternating or an N-group. Various related reduction theorems are also proved.