A generalized Hughes property of finite groups

A Hughes cover for exponent p (p a prime number) of a finite group is a union of subgroups whose (non-empty) complement consists of elements of order p. A proper Hughes subgroup is an instance of a Hughes cover; and its parent group is soluble by a well-known result of Hughes and Thompson. More generally an earlier result of the authors shows that a group with a Hughes cover of fewer than p subgroups is soluble. This article treats the insoluble groups having a Hughes cover for exponent p with exactly p subgroups: the almost simple groups with this property form a restricted class of projective special linear groups.