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  • Print publication year: 2003
  • Online publication date: January 2010

Some results on finite factorized groups

    • By A. Ballester-Bolinches, Departament d'Àlgebra, Universitat de València, C/ Doctor Moliner 50, 46100 Burjassot (València), Spain, John Cossey, Mathematics Department, School of Mathematical Sciences, The Australian National University, Canberra, 0200, Australia, Xiuyun Guo, Department of Mathematics, Shanxi University, Taiyuan, Shanxi 030006, People's Republic of China, 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 first 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
  • Publisher: Cambridge University Press
  • DOI: https://doi.org/10.1017/CBO9780511542770.007
  • pp 27-30

Summary

The well-known fact that a product of two normal supersoluble subgroups of a group is not necessarily supersoluble shows that the saturated formation of supersoluble groups need not be closed under the product of normal subgroups. This makes interesting the study of factorized groups whose subgroup factors are connected by certain permutability properties. Baer (see [2]) proved that if a group G is the product of two normal supersoluble subgroups, then G is supersoluble if and only if the commutator subgroup of G is nilpotent. This result has been generalized by Asaad and Shaalan ([1]) in the following sense: If G is the product of two subgroups H and K such that H permutes with every subgroup of K and K permutes with every subgroup of H, that is, G is the mutually permutable product of H and K, and G', the commutator subgroup of G is nilpotent, then G is supersoluble. Moreover they prove that in the case G = HK such that every subgroup of H permutes with every subgroup of K, that is, G is the totally permutable product of H and K, then if the factors H and K are supersoluble the group G is also supersoluble.

Further studies have been done by several authors within the framework of formation theory.