On the supersolubility of a finite group with NS-supplemented subgroups (Q2300120)

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 \(p \in \pi(B)\), there exists a Sylow \(p\)-subgroup \(B_{p}\) of \(B\) such that \(XB_{p}=B_{p}X\). The authors prove the following theorems.\par Let \(P\) be a Sylow \(p\)-subgroup of \(G\). If \(P\) is NS-supplemented in \(G\), then \(G\) is \(p\)-supersoluble if \(p \neq 3\) or if \(p = 3\) and \(G\) is 3-soluble.\par Let \(G\) be a \(p\)-soluble group and let \(P\) be a Sylow \(p\)-subgroup of \(G\). If for every maximal subgroup \(P_{i}\) of \(P\) and every \(q \in \pi(G)\), \(q \neq p\) there exists a Sylow \(q\)-subgroup \(Q\) of \(G\) such that \(QP_{i}=P_{i}Q\), then \(G\) is \(p\)-supersoluble.\par If all maximal subgroups of \(G\) are NS -supplemented in \(G\), then \(G\) is soluble.\par Finally, it is easy to use these theorems to show that if every non-cyclic Sylow subgroup is NS-supplemented in the group \(G\) or if for a soluble group \(G\), all maximal subgroups of any non-cyclic Sylow subgroup are NS-supplemented in the group, then \(G\) is supersoluble.