Abstract
In this note, we use fewer [Formula: see text]-subgroups [Formula: see text] with the condition [Formula: see text] to investigate the structure of finite groups. We prove that for a fixed prime [Formula: see text], a given Sylow [Formula: see text]-subgroup [Formula: see text] of a finite group [Formula: see text], and a power [Formula: see text] of [Formula: see text] dividing [Formula: see text] such that [Formula: see text], if [Formula: see text] is normal in [Formula: see text] for all non-cyclic normal subgroups [Formula: see text] of [Formula: see text] with [Formula: see text], then either [Formula: see text] is [Formula: see text]-supersoluble or else [Formula: see text]. This extends the main result of Guo and Isaacs (Conditions on [Formula: see text]-subgroups implying [Formula: see text]-nilpotence or [Formula: see text]-supersolvability, Arch. Math. 105 (2015) 215–222). We also derive some applications of the above result which extend some known results.
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