| Let N be a fixed normal subgroup of a group G and theta an irreducible character of N which is fixed by the conjugation action of G. Let chi be an irreducible character of G that restricts to a multiple of theta on N. Then d = chi(1)/theta(1) is an integer which behaves like an irreducible character degree of G/N. That is, d divides |G/N| and |G/N| ≥ d2. We can thus write |G/N| = d(d + e) for a non-negative integer e.;If N = 1 then d is an irreducible character degree of the group G and | G| = d(d + e). In this case, Berkovich has shown that if e = 1 then G is either the group of order 2 or is a 2-transitive Frobenius group. For e > 1, Snyder shows that d is bounded by a function of e. This bound is later improved by Isaacs and then by Durfee and Jensen.;In this paper, we discuss the generalized version of the problem in the special case that G/N is solvable. The results are similar to the N = 1 case for e = 1 but there is no longer a bound for d in terms of e for e > 1. We will show that if e = 1 then G/N is either the group or order 2 or a 2-transitive Frobenius group. We will also present examples that show that for any e value there is no bound on the possible d values. There are however restriction on the possible d values. In particular, for e > 0 if d > ( e--1)2 then e divides d and d/e + 1 is a prime power. Also for e > 0 if d > e5-- e then there are groups X,Y between N and G such that Y/X has order (d/e)(d/e + 1) and is either the group of order 2 or a 2-transitive Frobenius group. In addition to these results, we look at some special cases and discuss examples found using MAGMA and questions for future research. |
