Finite groups with certain subgroups of Sylow subgroups complemented. (Q965179)

Let \(\mathcal I\) be a saturated formation containing the class of supersoluble groups, \(G\) be a finite group with a normal subgroup \(E\) such that \(G/E\in\mathcal I\), and \(F^*(E)\) the generalised Fitting subgroup of \(E\). Theorem 1.3: If \(P\) is a Sylow subgroup of \(E\) and \(P\) has a proper subgroup \(D\) such that each subgroup \(H\) of \(P\) which is satisfying \({|H|\over|D|}= 1\) or \({|H|\over|D|}=\) a prime is complemented in \(G\), then \(G\in\mathcal I\). If \(P\) is a Sylow subgroup of \(F^*(E)\) and \(P\) has a proper subgroup \(D\) such that each subgroup \(H\) of \(P\) which is satisfying \({|H|\over|D|}= 1\) or \({|H|\over|D|}=\) a prime is complemented in \(G\), then \(G\in\mathcal I\) (Theorem 1.4).