Local Covering Subgroups in Finite Groups

A subgroup A of a finite group G is called a local covering subgroup of G if A~G=AB for all maximal G-invariant subgroup B of A~G=(A~g,g∈G).Let p be a prime and d be a positive integer.Assume that all subgroups of p~d,and all cyclic subgroups of order 4 when p~d=2 and a Sylow2-subgroup of G is nonabelian,of G are local covering subgroups.Then G is p-supersolvable whenever p~d=p or p~d≤(|G|)or p~d≤|O(G)|/p.