Abstract
Let [Formula: see text] be a finite solvable group, let Irr[Formula: see text] be the set of all irreducible monomial characters of [Formula: see text] and let [Formula: see text] be a prime. We prove that if [Formula: see text] for every nonlinear [Formula: see text][Formula: see text][Formula: see text], then [Formula: see text] has a normal [Formula: see text]-complement, and if [Formula: see text] is relatively prime to [Formula: see text] for every [Formula: see text], then [Formula: see text] has a normal Sylow [Formula: see text]-subgroup.
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