Permutable subnormal subgroups of finite groups. (Q2390922)

The article under review is devoted to the investigation of finite groups whose certain subnormal subgroups are permutable. A subgroup \(K\) of a group \(G\) is said to be `permutable' in \(G\) provided \(HK=KH\) for every subgroup \(H\) of \(G\). If permutability in \(G\) is a transitive relation, then \(G\) is called `\(\mathcal{PT}\)-group'. A subgroup \(H\) of \(G\) is said to be `permutable sensitive' in \(G\) if \[ \{N\mid N\text{ is permutable in }H\}=\{H\cap W\mid W\text{ is permutable in }G\}. \] A subgroup \(H\) is `conjugate-permutable' in \(G\) if \(HH^g=H^gH\), \(\forall g\in G\). The main results obtained in this article are the following theorems: Theorem A. For a \(p\)-group \(G\) the following statements are equivalent: (i) \(G\) has modular subgroup lattice; (ii) \(G\) has all subnormal subgroups of defect two permutable; (iii) \(G\) has all normal subgroups permutable sensitive; (iv) \(G\) has all conjugate-permutable subgroups permutable. Theorem B. The following statements are equivalent for a solvable group \(G\): (i) \(G\) is a \(\mathcal{PT}\)-group; (ii) every subnormal subgroup of defect two in \(G\) is permutable in \(G\); (iii) every normal subgroup of \(G\) is permutable sensitive in \(G\); (iv) every conjugate-permutable subgroup of \(G\) is permutable in \(G\). Finally, an example of a group with all subnormal subgroups of defect two permutable which is not a \(\mathcal{PT}\)-group is given.