Characterization of finite groups with some \(S\)-quasinormal subgroups of fixed order

Abstract

Let \(G\) be a finite group. A subgroup of \(G\) is said to be \(S\)-quasinormal in \(G\) if it permutes with every Sylow subgroup of \(G\). We fix in every non-cyclic Sylow subgroup \(P\) of the generalized Fitting subgroup a subgroup \(D\) such that \(1 < |D| < |P|\) and characterize \(G\) under the assumption that all subgroups \(H\) of \(P\) with \(|H| = |D|\) are \(S\)-quasinormal in \(G\). Some recent results are generalized.