Abstract
Let [Formula: see text] be a maximal subgroup of an [Formula: see text]-group [Formula: see text] with odd index and let [Formula: see text] be primitive. Lewis proved in this situation that [Formula: see text] divides [Formula: see text], and Isaacs and Wilde further refined this result by showing that either [Formula: see text] or [Formula: see text]. In this paper, we present an independent and simpler proof for these remarkable results and thereby obtain more detailed information regarding the structure of the group [Formula: see text] and the primitive character [Formula: see text]. In particular, [Formula: see text] is strongly irreducible in the sense of Brauer.
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