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Permutable subnormal subgroups of finite groups |
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| Authors: | A Ballester-Bolinches J C Beidleman John Cossey R Esteban-Romero M F Ragland Jack Schmidt |
| Institution: | 1. Departament d’ àlgebra, Universitat de València, Dr. Moliner, 50, E-46100, Burjassot, València, Spain 2. Department of Mathematics, 745 Patterson Office Tower, University of Kentucky, Lexington, KY, 40506-0027, U.S.A. 3. Mathematics Department, Mathematical Sciences Institute, Australian National University, 0200, Canberra, Australia 4. Institut Universitari de Matemàtica Pura i Aplicada, Universitat Politècnica de València, Camí de Vera, s/n, E-46022, València, Spain 5. Department of Mathematics, Auburn University Montgomery, P.O. Box 244023, Montgomery, AL, 36124-4023, U.S.A. |
| Abstract: | The aim of this paper is to prove certain characterization theorems for groups in which permutability is a transitive relation,
the so called  -groups. In particular, it is shown that the finite solvable  -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  -groups but all subnormal subgroups of defect two are permutable.
Received: 13 August 2008 |
| Keywords: | " target="_blank"> Permutable subnormal |
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