On groups satisfying the double chain condition on nonascendant subgroups

Document Type : Ischia Group Theory 2022

Author

Pingxiang Universiry, Pingxiang, China

Abstract

If $\theta$ is a subgroup property, a group $G$ is said to satisfy the double chain condition on $\theta$-subgroups if it admits no infinite double chain $$\cdots<X_{-n}<\cdots<X_{-1}<X_0<X_1<\cdots<X_n<\cdots$$ consisting of $\theta$-subgroups. Here we want to describe the structure of locally finite and locally nilpotent groups satisfying the double chain condition on non-ascendant subgroups in term of chain conditions and of ascendant subgroups.

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References

[1] M. Brescia, Groups satisfying the double chain condition on non-subnormal subgroups, J. Group Theory, 24 (2021) 95–108.
[2] M. Brescia, F C-groups with few subnormal non-normal subgroups, Mediterr. J. Math., 19 (2022) 19 pp.
[3] M. Brescia and F. de Giovanni, Groups satisfying the double chain condition on subnormal subgroups, Ric. Mat., 65 (2016) 255–261.
[4] M. Brescia and F. de Giovanni, Groups satisfying the double chain condition on non-pronormal subgroups, Riv. Mat. Univ. Parma, 8 (2017) 353–366.
[5] M. Brescia and F. de Giovanni, Groups satisfying the double chain condition on subnormal non-normal subgroups, Adv. Group Theory Appl., 13 (2022) 83–102.
[6] M. Brescia and A. Russo, Groups satisfying the double chain condition on abelian subgroups, Bull. Aust. Math. Soc., 99 (2019) 212–218.
[7] M. Brescia and A. Russo, Groups satisfying the double chain condition on non-abelian subgroups, J. Algebra Appl., 19 (2020) 14 pp.
[8] S. N. Chernikov, Infinite locally soluble groups, Mat. Sb., 7 (1940) 35–64.
[9] K. A. Hirsch, On infinite soluble groups I, Proc. London Math. Soc. (2), 44 (1938) 53–60.
[Malcev(1951)] A. I. Malcev, On certain classes of infinite soluble groups, Mat. Sb., 28 (1951) 567–588.
[10] B. I. Plotkin, On groups with finiteness conditions for abelian subgroups, Dokl. Akad. Nauk. SSSR, 107 (1956) 648–651.
[11] D. Robinson, Finiteness Conditions and Generalized Soluble Groups, Springer, Berlin, 1972.
[12] D. Robinson, A Course in the Theory of Groups, Springer, Berlin, 1982.
[13] O. J. Schmidt, Infinite soluble groups, Mat. Sb., 17 (1945) 145–162.
[14] T. S. Shores, A chain condition for groups, Rocky Mountain J. Math., 3 (1973) 83–89.
[15] V. P. SĖŒunkov, On locally finite groups with a minimality conditions for abelian subgroups,Algebra and Logic, 10 (1971) 199–225.
[16] D. I. Zaicev, On solvable subgroups of locally solvable groups, Soviet Math. Dokl., 15 (1974) 342–345.
International Journal of Group Theory
Volume 14, Issue 1 - Serial Number 1
Proceedings of the Ischia Group Theory 2022-Part 2
March 2025
Pages 1-7

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History

  • Receive Date: 18 July 2023
  • Revise Date: 23 September 2023
  • Accept Date: 29 September 2023
  • Published Online: 04 October 2023

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