Ballester-Bolinches, A.; Beidleman, J. C.; Cossey, John; Esteban-Romero, R.; Ragland, M. F.; Schmidt, Jack Permutable subnormal subgroups of finite groups. (English) Zbl 1182.20024 Arch. Math. 92, No. 6, 549-557 (2009). 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. Reviewer: Bui Xuan Hai (Ho Chi Minh City) Cited in 2 Documents MSC: 20D40 Products of subgroups of abstract finite groups 20D35 Subnormal subgroups of abstract finite groups 20D10 Finite solvable groups, theory of formations, Schunck classes, Fitting classes, \(\pi\)-length, ranks 20D15 Finite nilpotent groups, \(p\)-groups 20D30 Series and lattices of subgroups Keywords:permutable subgroups; subnormal subgroups; \(\mathcal{PT}\)-groups; conjugate-permutable subgroups; modular \(p\)-groups; transitive permutability Software:permut PDFBibTeX XMLCite \textit{A. Ballester-Bolinches} et al., Arch. Math. 92, No. 6, 549--557 (2009; Zbl 1182.20024) Full Text: DOI Link