Permutable subnormal subgroups of finite groups

The final publication is available at Springer via http://dx.doi.org/10.1007/s00013-009-2976-xThe aim of this paper is to prove certain characterization theorems for groups in which permutability is a transitive relation, the so called PT -groups. In particular, it is shown that the finite solvable PT -groups, the finite solvable groups in which every subnormal subgroup of defect two is permutable, the finite solvable groups in which every normal subgroup is permutable sensitive, and the finite solvable groups in which conjugatepermutability and permutability coincide are all one and the same class. This follows from our main result which says that the finite modular p-groups, p a prime, are those p-groups in which every subnormal subgroup of defect two is permutable or, equivalently, in which every normal subgroup is permutable sensitive. However, there exist finite insolvable groups which are not PT -groups but all subnormal subgroups of defect two are permutable.This paper was started and completed in two visits of the first author to the Department of Mathematics of the Australian National University, Canberra, Australia. He is grateful to the Department of Mathematics for its warm hospitality and financial support while this investigation was carried out. He also gratefully acknowledges financial support by Grant MTM2007-68010-C03-02 from MEC (Spanish Government) and FEDER. The research of the fourth author has been supported by the Grants MTM2007-68010-C03-02 from MEC (Spanish Government) and FEDER, and GV/2007/243 from Generalitat (Valencian Government). He thanks all these institutions for their financial support.Ballester Bolinches, A.; Beidleman, JC.; Cossey, J.; Esteban Romero, R.; Rangland, MF.; Schmidt, J. (2009). Permutable subnormal subgroups of finite groups. Archiv der Mathematik. 92(6):549-557. doi:10.1007/s00013-009-2976-xS54955792