Abstract
We obtain a complete characterization of the structure of a finite group [Formula: see text] in which every maximal subgroup is nilpotent or a TI-subgroup or has order [Formula: see text] for any fixed prime divisor [Formula: see text] of [Formula: see text]. Moreover, we show that there exists at most one prime divisor [Formula: see text] of [Formula: see text] such that [Formula: see text] is neither [Formula: see text]-nilpotent nor [Formula: see text]-closed.
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