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The influence of weakly-supplemented subgroups on the structure of finite groups

Let G be a ﬁnite group. A subgroup L of G is said to be weakly-supplemented in G if there exists a proper subgroup M of G such that G = LM . The key of this paper is to determine the supersolvability and p -nilpotency of ﬁnite groups under assumption that some primary subgroups are weakly-supplemented.