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Permutability and subnormality in finite groups
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- By Manuel J. Alejandre, Centro de Investigación Operativa, Universidad Miguel Hernández, Avda. del Ferrocarril s/n, 03202 Elche, Spain, A. Ballester-Bolinches, Departament d'Àlgebra, Universitat de València, C/ Doctor Moliner 50, 46100 Burjassot (València), Spain, R. Esteban-Romero, Departamento de Matemática Aplicada, E.U.I., Universidad Politécnica de Valencia, Camino de Vera, s/n, 46071 Valencia, Spain, M. C. Pedraza-Aguilera, Departamento de Matemática Aplicada, E.U.I., Universidad Politécnica de Valencia, Camino de Vera, s/n, 46071 Valencia, Spain; The work of the second and last authors is supported by Proyecto PB97-0674-C02-02 of DGICYT, MEC, Spain
- Edited by C. M. Campbell, University of St Andrews, Scotland, E. F. Robertson, University of St Andrews, Scotland, G. C. Smith, University of Bath
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- Book:
- Groups St Andrews 2001 in Oxford
- Published online:
- 11 January 2010
- Print publication:
- 06 November 2003, pp 1-5
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- Chapter
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Summary
Notation and terminology
A group G is said to be a T-group if every subnormal subgroup of G is normal in G, that is, if normality is a transitive relation. These groups have been widely studied (see [10], [11], or [14]).
A subgroup H of a group G is said to be permutable (or quasinormal) in G if HK = KH for all subgroups K of G. Permutability can be considered thus as a weak form of normality. The study of groups G in which permutability is transitive, that is, H permutable in K and K permutable in G always imply that H is permutable in G, has been a successful field of research in recent years. Such groups are called PT-groups. According to a theorem of Kegel [12, Satz 1], every permutable subgroup of G is subnormal in G. Consequently, PT-groups are exactly those groups in which subnormality and permutability coincide; that is, those groups in which every subnormal subgroup permutes with every other subgroup. Therefore, every T-group is clearly a PT-group.
One could wonder what would happen if we did not require that every subnormal subgroup of a group G permutes with any other subgroup of G, but only with a certain family of its subgroups. In this direction, those groups in which every subnormal subgroup of G permutes with every Sylow p-subgroup of G for each prime p have sometimes been called T*-groups (see [3]) or also (π – q)-groups (see [1]).