On Π-Normal Subgroups and the p-Supersolvability of Finite Groups

Rather than using all maximal subgroups of a Sylow p-subgroup, we work with a prescribed subfamily. For a p-solvable group G, we show that p-supersolvability follows when the members of this subfamily are Π-normal in the Sylow normalizer and when G contains a further Π-normal subgroup lying between the derived subgroup of the Sylow p-subgroup and the Frattini subgroup of its normalizer. This result replaces some stronger embedding hypotheses occurring in previous local criteria and recovers several earlier conclusions under the corresponding special assumptions. As applications, analogous criteria are obtained for p-nilpotency and supersolvability. The approach shows that the structure of the group can, under suitable conditions, be determined from information attached to only a limited number of local subgroups.