On the supersolubility of a finite group with NS-supplemented subgroups.

A subgroup A of a finite group G is said to be NS-supplemented in G, if there exists a subgroup B of G such that G=AB and whenever X is a normal subgroup of A and p ∈ π (B) , there exists a Sylow p-subgroup Bp of B such that X B p = B p X . In this paper, we prove the supersolubility of a group G in the following cases: every non-cyclic Sylow subgroup of G is NS-supplemented in G; G is soluble and all maximal subgroups of every non-cylic Sylow subgroup of G are NS-supplemented in G. The solubility of a group with NS-supplemented maximal subgroups is obtained. [ABSTRACT FROM AUTHOR]