Degree divisibility in character correspondences. (Q863362)

This paper continues the author's work on character correspondences in finite solvable groups. In [\textit{A. Turull}, J. Algebra 295, No. 1, 157-178 (2006; Zbl 1107.20008), and ``Odd character correspondences in solvable groups'', J. Algebra (to appear)] the author proved some very strong generalizations of the well-known McKay Conjecture in the case of solvable groups, i.e., he established a bijection \(f\) between the irreducible \(p'\)-characters of \(G\) and those of a normalizer \(N\) of a Sylow \(p\)-subgroup of \(G\) that enjoys many additional properties. In the paper under review, he proves that this \(f\) can even be chosen such that \(f(\chi)(1)\) divides \(\chi(1)\) for any irreducible \(p'\)-character \(\chi\), more precisely that \(\chi(1)/f(\chi)(1)\) divides \(|G:N|\). He also shows that such a precise bijection \(f\) cannot always exist for non-solvable groups.