Some generalizations of Shao and Beltrán’s theorem
Let [Formula: see text] and [Formula: see text] be finite groups of relative coprime orders and [Formula: see text] act on [Formula: see text] via automorphisms. In this paper, we prove that when every maximal [Formula: see text]-invariant subgroup of [Formula: see text] that contains the normalizer of some Sylow subgroup has prime index, then [Formula: see text] is supersolvable; if every non-nilpotent maximal [Formula: see text]-invariant subgroup of [Formula: see text] has prime index or is normal in [Formula: see text], then [Formula: see text] is a Sylow tower group.
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