Properties of Abelian-by-cyclic shared by soluble finitely generated groups

Properties of Abelian-by-cyclic shared by soluble finitely generated groups

Our main result states that if G is a finitely generated soluble group having a normal Abelian subgroup A, such that G/A and ⟨x, a⟩ are nilpotent (respectively, finite-by-nilpotent, periodic-by-nilpotent, nilpotent-by-finite, finite-by-supersoluble, supersoluble-by-finite) for all (x, a) ∈ G × A, then so is G. We deduce that if X is a subgroup and quotient closed class of groups and if all 2-generated Abelian-by-cyclic groups of X are nilpotent (respectively, finite-by-nilpotent, periodic-by-nilpotent, nilpotent-by-finite, finite-by-supersoluble, supersoluble-by-finite), then so are all finitely generated soluble groups of X. We give examples that show that our main result is not true for other classes of groups, like the classes of Abelian, supersoluble, and F C -groups.

___

  • [1] Bray HG, Deskins WE, Johnson D. Between Nilpotent and Solvable. Poly-gonal, Washington, 1982.
  • [2] Brookes CJB. Engel elements of soluble groups. Bulletin of the London Mathematical Society 1986; 18 (1): 7-10. doi: 10.1112/blms/18.1.7
  • [3] Endimioni G, Traustason G. On torsion-by-nilpotent groups. Journal of Algebra 2001; 241 (2): 669-676. doi: 10.1006/jabr.2001.8772
  • [4] Groves JRJ. Soluble groups in which every finitely generated subgroup is finitely presented. Journal of the Australian Mathematical Society (Series A) 1978; 26 (1): 115-125. doi: 10.1017/S1446788700011599
  • [5] Groves JRJ. A conjecture of Lennox and Wiegold concerning supersoluble groups. Journal of the Australian Mathematical Society (Series A) 1983; 35 (2): 218-220. doi: 10.1017/S1446788700025702
  • [6] Lennox JC, Wiegold J. Extensions of a problem of Paul Erdős on groups. Journal of the Australian Mathematical Society (Series A) 1981; 31 (4): 459-463. doi: 10.1017/S1446788700024253
  • [7] Lennox JC. On a centrality property of finitely generated torsion free soluble groups. Journal of Algebra 1971; 18 (4): 541–548. doi: 10.1016/0021-8693(71)90137-2
  • [8] Robinson DJS. A course in the theory of groups. Springer-Verlag, New York, 1982.
  • [9] Robinson DJS. Finiteness Conditions and Generalized Soluble Groups Part 2. Springer-Verlag, New York, 1972.
  • [10] Robinson DJS. Finiteness Conditions and Generalized Soluble Groups Part 1. Springer-Verlag, New York, 1972.
  • [11] Trabelsi N. Soluble groups with a condition on infinite subsets. Algebra Colloquium 2002; 9 (4): 427-432.
  • [12] Wall CTC. Poincaré complexes I. Annals of Mathematics (Second Series) 1967; 86 (2): 213-245. doi: 10.2307/1970688