Primitive characters of maximal subgroups of solvable groups

Let G be an M-group, and suppose that H is a maximal subgroup of odd index and that beta is an element of Irr H is primitive. It was shown by Mark Lewis that beta(1)(2) divides vertical bar G : H vertical bar. We show in this situation that either beta(1)(2) = vertical bar G : H vertical bar or beta(1) = 1. A number of related results are established for groups that are not necessarily M-groups. (C) 2009 Elsevier Inc. All rights reserved.