On quasi-primitive characters of solvable groups

Abstract

Let [Formula: see text] be a quasi-primitive character with odd degree, and suppose that [Formula: see text] is a [Formula: see text]-solvable group. Wilde associated to [Formula: see text] a unique conjugacy class of subgroups [Formula: see text] satisfying [Formula: see text]. We construct in this situation a sequence of character pairs [Formula: see text], where [Formula: see text] is quasi-primitive and each [Formula: see text] is uniquely determined up to conjugacy in [Formula: see text], such that [Formula: see text] and [Formula: see text]. Furthermore, we have [Formula: see text] for each [Formula: see text], and in particular [Formula: see text]. We also prove that the subgroups [Formula: see text] and [Formula: see text] are conjugate in [Formula: see text], and thus present a new description for Wilde’s result.