Remote Access St. Petersburg Mathematical Journal

St. Petersburg Mathematical Journal

ISSN 1547-7371(online) ISSN 1061-0022(print)

Request Permissions   Purchase Content 
 
 

 

Free subgroups in almost subnormal subgroups of general skew linear groups


Authors: N. K. Ngoc, M. H. Bien and B. X. Hai
Original publication: Algebra i Analiz, tom 28 (2016), nomer 5.
Journal: St. Petersburg Math. J. 28 (2017), 707-717
MSC (2010): Primary 20G15
DOI: https://doi.org/10.1090/spmj/1468
Published electronically: July 25, 2017
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Let $ D$ be a weakly locally finite division ring and $ n$ a positive integer. The problem under study concerns the existence of noncyclic free subgroups in noncentral almost subnormal subgroups of the general linear group $ \operatorname {GL} _n(D)$. Further, some applications are also investigated. In particular, all infinite finitely generated almost subnormal subgroups of $ \operatorname {GL} _n(D)$ are described.


References [Enhancements On Off] (What's this?)

  • [1] E. Artin, Geometric algebra, Intersci. Publ., Inc., New York-London, 1957. MR 0082463
  • [2] S. Bachmuth, Problem section, Proc. Second Intern. Conf. Theory of Groups (Canberra, Australia, 1973), Lecture Notes in Math., vol. 372, Springer-Verlag, Berlin, 1974, pp. 736. MR 0364402
  • [3] M. H. Bien, On some subgroups of $ D^*$ which satisfy a generalized group identity, Bull. Korean Math. Soc. 52 (2015), no. 4, 1353-1363. MR 3385773
  • [4] M. H. Bien and D. H. Dung, On normal subgroups of division rings which are radical over a proper division subring, Studia Sci. Math. Hungar. 51 (2014), no. 2, 231-242. MR 3238133
  • [5] M. H. Bien, D. Kiani, and M. Ramezan-Nassab, Some skew linear groups satisfying generalized group identities, Comm. Algebra 44 (2016), 2362-2367; DOI:10.1080/00927872.2015.1044109. MR 3492161
  • [6] J. P. Bell, V. Drensky, and Y. Sharifi, Shirshov's theorem and division rings that are left algebraic over a subfield, J. Pure Appl. Algebra 217 (2013), no. 9, 1605-1610. MR 3042623
  • [7] K. Chiba, Free subgroups and free subsemigroups of division rings, J. Algebra 184 (1996), no. 2, 570-574. MR 1409229
  • [8] T. T. Deo, M. H. Bien, and B. X. Hai, On radicality of maximal subgroups in $ {\mathrm {GL}}_n(D)$, J. Algebra 365 (2012), 42-49. MR 2928452
  • [9] I. Z. Golubchik and A. V. Mikhalev, Generalaized group identities in classical groups, Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 114 (1982), 96-119; English transl., J. Soviet Math. 27 (1984), no. 4, 2904-2918. MR 669562
  • [10] J. Z. Gonçalves, Free subgroups of units in group rings, Canad. Math. Bull. 27 (1984), no. 3, 309-312. MR 749637
  • [11] -, Free groups in subnormal subgroups and the residual nilpotence of the group of units of group rings, Canad. Math. Bull. 27 (1984), no. 3, 365-370. MR 749646
  • [12] J. Z. Gonçalves and A. Mandel, Are there free groups in division rings? Israel J. Math. 53 (1986), no. 1, 69-80. MR 861898
  • [13] J. Z. Gonçalves and D. S. Passman, Free groups in normal subgroups of the multiplicative group of a division ring, J. Algebra 440 (2015), 128-144. MR 3373391
  • [14] J. Z. Gonçalves and M. Shirvani, Algebraic elements as free factors in simple Artinian rings, Contemp. Math., vol. 499, Amer. Math. Soc., Providence, RI, 2009, pp. 121-125. MR 2581930
  • [15] B. X. Hai, T. T. Deo, and M. H. Bien, On subgroups of division rings of type $ 2$, Studia Sci. Math. Hungar. 49 (2012), no. 4, 549-557. MR 3098299
  • [16] B. X. Hai and N. K. Ngoc, A note on the existence of non-cyclic free subgroups in division rings, Arch. Math. (Basel) 101 (2013), 437-443. MR 3125560
  • [17] B. X. Hai, M. H. Bien, and T. T. Deo, On the Gelfand-Kirillov dimension of weakly locally finite division rings, http://arxiv.org/abs/1510.08711.
  • [18] B. Hartley, Free groups in normal subgroups of unit groups and arithmetic groups, Contemp. Math., vol. 93, Amer. Math. Soc., Providence, RI, 1989, pp. 173-177. MR 1003352
  • [19] R. Hazrat and A. R. Wadsworth, On maximal subgroups of the multiplicative group of a division algebra, J. Algebra 322 (2009), no. 7, 2528-2543. MR 2553693
  • [20] R. Hazrat, M. Mahdavi-Hezavehi, and M. Motiee, Multiplicative groups of division rings, Math. Proc. R. Ir. Acad. 114A (2014), no. 1, 37-114. MR 3354612
  • [21] T. Y. Lam, A first course in noncommutative rings, Grad. Texts. Math., vol. 131, Springer-Verlag, New York, 1991. MR 1125071
  • [22] A. I. Lichtman, On subgroups of the multiplicative group of skew fields, Proc. Amer. Math. Soc. 63 (1977), no. 1, 15-16. MR 0477432 (56:5744)
  • [23] -, Free subgroups of normal subgroups of the multiplicative group of skew fields, Proc. Amer. Math. Soc. 71 (1978), no. 2, 174-178. MR 0480623
  • [24] -, Free subgroups in linear groups over some skew fields, J. Algebra 105 (1987), no. 1, 1-28. MR 871744
  • [25] M. Mahdavi-Hezavehi, M. G. Mahmudi, and M. G. S. Yasamin, Finitely generated subnormal subgroups of $ {\mathrm {GL}}_n(D)$ are central, J. Algebra 225 (2000), no. 2, 517-521. MR 1741550
  • [26] J. Tits, Free subgroups in linear groups, J. Algebra 20 (1972), 250-270. MR 0286898
  • [27] G. M. Tomanov, Generalized group identities in linear groups, Mat. Sb. (N.S.) 123 (1984), no. 1, 35-49; English transl., Math. USSR-Sb 51 (1985), no. 1, 33-46. MR 728928
  • [28] Z. Reichstein and N. Vonessen, Free subgroups of division algebras, Comm. Algebra 23 (1995), no. 6, 2181-2185. MR 1327133
  • [29] L. H. Rowen, Polynomial identities in ring theory, Pure Appl. Math., vol. 84, Acad. Press, New York, 1980. MR 576061
  • [30] M. Shirvani and B. A. F. Wehrfritz, Skew linear groups, London Math. Soc. Lecture Note Ser., vol. 118, Cambridge Univ. Press, Cambridge, 1986. MR 883801

Similar Articles

Retrieve articles in St. Petersburg Mathematical Journal with MSC (2010): 20G15

Retrieve articles in all journals with MSC (2010): 20G15


Additional Information

N. K. Ngoc
Affiliation: Faculty of Mathematics and Computer Science, University of Science, VNU-HCM, 227 Nguyen Van Cu Str., Dist. 5, HCM-City, Vietnam
Email: nkngoc1985@gmail.com

M. H. Bien
Affiliation: Department of Basic Sciences, University of Architecture, 196 Pasteur Str., Dist. 1, HCM-City, Vietnam
Email: maihoangbien012@yahoo.com

B. X. Hai
Affiliation: Faculty of Mathematics and Computer Science, University of Science, VNU-HCM, 227 Nguyen Van Cu Str., Dist. 5, HCM-City, Vietnam
Email: bxhai@hcmus.edu.vn

DOI: https://doi.org/10.1090/spmj/1468
Keywords: Division rings, linear groups, almost subnormal subgroups, noncyclic free subgroups, generalized group identity
Received by editor(s): April 20, 2016
Published electronically: July 25, 2017
Additional Notes: This research was funded by Vietnam National University HoChiMinh City (VNU-HCM) under grant no. B2016-18-01. The authors thank the referee for his/her comments
Article copyright: © Copyright 2017 American Mathematical Society

American Mathematical Society