Some results on supersolvability of finite groups. (Q938326)

If \(\Sigma\) is a Sylow system of the finite group \(G\), then a subgroup \(H\) of \(G\) is called \(\Sigma\)-quasinormal if \(H\) permutes with every element of \(\Sigma\). Theorem 1.1: \(G\) is supersoluble if and only if there is a normal subgroup \(H\) of \(G\) such that \(G/H\) is supersoluble and there exists a Sylow system \(\Sigma\) of \(G\) such that for every \(P_i\in\Sigma\), the subgroup \(P_i\cap H\) is the product of cyclic \(\Sigma\)-quasinormal subgroups. Let \(F^*(G)\) the generalised Fitting subgroup of \(G\) and \(G'\) the commutator subgroup of \(G\). Theorem 1.2: \(G\) is supersoluble if and only if there is a Sylow system \(\Sigma\) of \(G\) such that, for every \(P_i\in\Sigma\), one can write \(F^*(G)\cap P_i\) as a product of \(\Sigma\)-quasinormal cyclic subgroups and \(G'\leq F^*(G)\). One further result studying factorised groups, Theorem 1.3, extends to saturated formations containing the class of supersoluble groups.