On $Π$-property of $p$-subgroups and the $p$-supersolvability of Finite Groups (European Journal of Pure and Applied Mathematics, 2026)
A subgroup $H$ of a group $G$ is said to satisfy $Π$-property in $G$ such that for any $G$-chief factor $U/V$, $|G/V:N_{G/V}(HV/V\cap U/V)|$ is a $π(HV/V\cap U/V)$-number. In this paper, we present a new criterion for $p$-supersolvability of finite groups by using of a small quantity of maximal subgroups of a Sylow $p$-subgroup satisfying the $Π$-property. As applications, we obtain some sufficient conditions for a finite group to be $p$-nilpotent and supersolvable. A number of known results are improved and extended.
Keywords: Finite group, $Π$-property, $p$-supersolvable