On Π-supplemented subgroups of a finite group

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A subgroup H of a finite group G is said to satisfy Π-property in G if for every chief factor L/K of G, |G/K:NG/ₖ(HK/K∩ L/K)| is a π(HK/K∩ L/K)-number. A subgroup H of G is called to be Π-supplemented in G if there exists a subgroup T of G such that G=HT and H∩ T≤ I≤ H, where I satisfies Π-property in G. In this paper, we investigate the structure of a finite group G under the assumption that some primary subgroups of G are Π-supplemented in G. The main result we proved improves a large number of earlier results.