Groups with a character of large degree relative to a normal subgroup

Let N be a normal subgroup of a finite group G and theta be 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 divides vertical bar G : N vertical bar and has vertical bar G : N vertical bar >= d(2). We can thus write vertical bar G/N vertical bar = d(d+ e) for a non-negative integer e and ask what can be said about d and G/N for a given e. This is a generalization of a problem considered by Snyder [12] where he takes d to an irreducible character degree of G and writes vertical bar G vertical bar = d(d + e). Berkovich has shown in [1] that if e = 1, then G is a sharply 2-transitive group. For e not equal 1, Snyder shows in [12] that d is bounded by a function of e. This bound is later improved by Isaacs in [7] and then by Durfee and Jensen in [2] and Lewis in [9]. In this more general version of the problem, we will work under the assumption that G/N is solvable. We will show that for e > 0, if d > (e - 1)(2) then e divides d and d/e +1 is a prime power. If in addition, either d > e(5) - e, (d/e,e) = 1, or (d/e + 1, e) = 1 then there exist groups X, Y with N subset of X Y subset of G such that Y/X is a sharply 2-transitive group of order (d/e)(d/e+1). (C) 2014 Elsevier Inc. All rights reserved.