Let $FL_{\nu}(K)$ be a finitary linear group of degree $\nu$ over a ring $K$, and let $K$ be an associative ring with the unit. We study periodic subgroups of $FL_{\nu}(K)$ in the cases when $K$ is an integral ring (Theorem 1) and a commutative Noetherian ring (Theorem 2). In both cases we prove that the periodic subgroups of $FL_{\nu}(K)$ are locally finite and describe their normal structure. In Theorem 3 we describe the structure of finitely generated solvable subgroups of $FL_{\nu}(K)$ if $K$ is an integral ring, a commutative Noetherian ring, or an arbitrary commutative ring. We show that this structure is most complicated in the latter case.
Keywords: finitary linear group, commutative Noetherian ring, locally finite group.
The paper was received by the Editorial Office on September 20, 2017
Olga Yurievna Dashkova, Dr. Phys.-Math. Sci, Prof., the Branch of Moscow State University named
after M.V. Lomonosov in Sevastopol, 299001 Russia, e-mail: odashkova@yandex.ru .
Mohamed Ahmed Salim, Dr. Phys.-Math. Sci, Prof., United Arab Emirates University, Al-Ain,
15551 United Arab Emirates, e-mail: MSalim@uaeu.ac.ae .
Olga Alekseevna Shpyrko, Cand. Phys.-Math. Sci, Associate Prof., the Branch of Moscow State
University named after M.V. Lomonosov in Sevastopol, 299001 Russia, e-mail: shpyrko@mail.ru .
REFERENCES
1. Dixon M.R., Kurdachenko L.A., Otal J. On the structure of some infinite dimensional linear groups. Commun. Algebra, 2017, vol. 45, no. 1, pp. 234-246. doi: 10.1080/00927872.2016.1175593.
2. Kurdachenko L.A., Munoz-Escolano J.M., Otal J. Antifinitary linear groups. Forum Math., 2008, vol. 20, no. 1, pp. 27-44. doi:10.1515/FORUM.2008.002.
3. Kurdachenko L.A., Munoz-Escolano J.M., Otal J. Locally nilpotent linear groups with the weak chain conditions on subgroups of infinite central dimension. Publ. Mat., 2008, vol. 52, no. 1, pp. 151-169.
4. Dashkova O.Yu., Dixon M.R., Kurdachenko L.A. Linear groups with rank restrictions on the subgroups of infinite central dimension. J. Pure Appl. Algebra, 2007, vol. 208, no. 3, pp. 785-795. doi: 10.1016/j.jpaa.2006.04.002.
5. Dixon M.R., Evans M.J., Kurdachenko L.A. Linear groups with the minimal condition on subgroups of infinite central dimension. J. Algebra, 2004, vol. 277, no. 1, pp. 172-186. doi: 10.1016/j.jalgebra.2004.02.029.
6. Kurdachenko L.A., Subbotin I.Ya. Linear groups with the maximal condition on subgroups of infinite central dimension. Publ. Mat., 2006, vol. 50, no. 1, pp. 103-131.
7. Dashkova O. Yu. Solvable infinite-dimensional linear groups with restrictions on the nonabelian subgroups of infinite rank. Sib. Math. J., 2008, vol. 49, no. 6, pp. 1023-1033. doi: 10.1007/s11202-008-0098-5.
8. Dashkova O. Yu. Infinite-dimensional linear groups with restrictions on subgroups that are not soluble $A_3$-groups. Algebra Logic, 2007, vol. 46, no. 5, pp. 297–302. doi: 10.1007/s10469-007-0030-2.
9. Dashkova O. Yu. Locally soluble infinite-dimensional linear groups with restrictions on nonabelian subgroups of infinite ranks. Algebra Logic, 2008, vol. 47, no. 5, pp. 340-347. doi: 10.1007/s10469-008-9025-x.
10. Merzlyakov Yu. I. Equisubgroups of unitriangular groups: A criterion for selfnormalizability. Russ. Acad. Sci., Dokl., Math., 1995, vol. 50, no. 3, pp. 507-511.
11. Levchuk V.M. Some locally nilpotent rings and their adjoined groups. Math. Notes, 1987, vol. 42, no. 5, pp. 848-853. doi:10.1007/BF01137426.
12. Wehrfritz B.A.F. Infinite linear groups. Berlin, Heidelberg, N. Y., Springer-Verlag, 1973, 229 p. ISBN: 3-540-06132-0.
13. Kargapolov M.I., Merzljakov Ju.I. Fundamentals of the theory of groups. Transl. from the 2nd Russian ed., N. Y., Heidelberg, Berlin, Springer-Verlag,1979, Ser. Graduate Texts in Math., vol. 62, 203 p. ISBN: 978-1-4612-9966-0. Original Russian text (1st ed.) published in M.I. Kargapolov, Yu.I. Merzlyakov Osnovy teorii grupp, Moscow, Nauka Publ., 1972, 240 p.
14. Kurosh A.G. Group theory. Moscow: Nauka, 1967, 648 p. Transl. from the 2nd Russian ed., N. Y., Chelsea Publ. Co., 1960, vol. 1, 272 p. ISBN: 978-0828401074, vol. 2: 308 p. ISBN: 978-0821834770.