TY - JOUR

T1 - Permutable subnormal subgroups of finite groups

AU - Ballester-Bolinches, A.

AU - Beidleman, J. C.

AU - Cossey, John

AU - Esteban-Romero, R.

AU - Ragland, M. F.

AU - Schmidt, Jack

PY - 2009/6

Y1 - 2009/6

N2 - The 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 conjugate-permutability 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.

AB - The 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 conjugate-permutability 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.

KW - Conjugate-permutable

KW - Modular p-group

KW - PT -group

KW - Permutable

KW - Subnormal

UR - http://www.scopus.com/inward/record.url?scp=67949088173&partnerID=8YFLogxK

U2 - 10.1007/s00013-009-2976-x

DO - 10.1007/s00013-009-2976-x

M3 - Article

SN - 0003-889X

VL - 92

SP - 549

EP - 557

JO - Archiv der Mathematik

JF - Archiv der Mathematik

IS - 6

ER -