Finite gropus whose proper subgroups of order divisible by $p$ are supersolvable

Abstract

Let $p$ be an odd prime, and let $G$ be a finite group whose order is divisible by $p$. Suppose that every proper subgroup of $G$ whose order is divisible by $p$ is supersolvable, while $G$ itself is not. In this paper, we investigate the arithmetic and structural properties of such groups, addressing both the solvable and nonsolvable cases.