Local Covering Subgroups in Finite Groups.

A subgroup A of a finite group G is called a local covering subgroup of G if AG = AB for all maximal G-invariant subgroup B of AG = 〈Ag, g ∊ G〉. Let p be a prime and d be a positive integer. Assume that all subgroups of pd, and all cyclic subgroups of order 4 when pd = 2 and a Sylow 2-subgroup of G is nonabelian, of G are local covering subgroups. Then G is p-supersolvable whenever pd = p or p d ≤ | G | p or pd ≤ ∣Op'p(G)∣p/p. [ABSTRACT FROM AUTHOR]