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On ℳ-supplemented subgroups of finite groups

Abstract A subgroup H is called ℳ-supplemented in a finite group G if there exists a subgroup B of G such that G = HB and such that H 1 B is a proper subgroup of G for any maximal subgroup H 1 of H. In this paper we fix a subgroup D in every non-cyclic Sylow subgroup P of G satisfying 1 < D < P and study the structure of G under the assumption that all subgroups H of P with |H| = |D| are ℳ-supplemented in G or have a supersolvable supplement in G.