Sylow permutable subnormal subgroups of finite groups (Q1851333)

Let \(p\) be a prime. The authors consider groups \(G\) with modular Sylow \(p\)-subgroup \(S\) and show for this class that \(p\)-nilpotency of \(G\) and \(N_G(S)\) imply each other. Also, \(p\)-subgroups that are permutable with all Sylow subgroups are permutable with all subgroups if \(S\) is modular. If all Sylow subgroups are modular, permutability with all Sylow subgroups implies permutability with all subgroups (Theorems 1 and 2). Implications are given concerning group classes near to PST-groups.