Free subgroups in almost subnormal subgroups of general skew linear groups

Ngoc, N.; Bien, M.; Hai, B.

doi:10.1090/spmj/1468

Free subgroups in almost subnormal subgroups of general skew linear groups
HTML articles powered by AMS MathViewer

by N. K. Ngoc, M. H. Bien and B. X. Hai

St. Petersburg Math. J. 28 (2017), 707-717

DOI: https://doi.org/10.1090/spmj/1468

Published electronically: July 25, 2017

PDF | Request permission

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

E. Artin, Geometric algebra, Interscience Publishers, Inc., New York-London, 1957. MR 0082463
John Cossey (ed.), Problem section, Proceedings of the Second International Conference on the Theory of Groups (Australian Nat. Univ., Canberra, 1973) Lecture Notes in Math., Vol. 372, Springer, Berlin, 1974, pp. 733–740. MR 0364402
Mai Hoang Bien, On some subgroups of $D^*$ which satisfy a generalized group identity, Bull. Korean Math. Soc. 52 (2015), no. 4, 1353–1363. MR 3385773, DOI 10.4134/BKMS.2015.52.4.1353
Mai Hoang Bien and Duong Hoang 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, DOI 10.1556/SScMath.51.2014.2.1265
M. H. Bien, D. Kiani, and M. Ramezan-Nassab, Some skew linear groups satisfying generalized group identities, Comm. Algebra 44 (2016), no. 6, 2362–2367. MR 3492161, DOI 10.1080/00927872.2015.1044109
Jason P. Bell, Vesselin Drensky, and Yaghoub 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, DOI 10.1016/j.jpaa.2012.11.015
Katsuo Chiba, Free subgroups and free subsemigroups of division rings, J. Algebra 184 (1996), no. 2, 570–574. MR 1409229, DOI 10.1006/jabr.1996.0275
Trinh Thanh Deo, Mai Hoang Bien, and Bui Xuan Hai, On the radicality of maximal subgroups in $\textrm {GL}_n(D)$, J. Algebra 365 (2012), 42–49. MR 2928452, DOI 10.1016/j.jalgebra.2012.04.027
I. Z. Golubchik and A. V. Mikhalëv, Generalized group identities in classical groups, Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 114 (1982), 96–119, 219 (Russian). Modules and algebraic groups. MR 669562
Jairo Zacarias Gonçalves, Free subgroups of units in group rings, Canad. Math. Bull. 27 (1984), no. 3, 309–312. MR 749637, DOI 10.4153/CMB-1984-046-7
Jairo Z. Gonçalves, 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, DOI 10.4153/CMB-1984-055-6
Jairo Z. Gonçalves and Arnaldo Mandel, Are there free groups in division rings?, Israel J. Math. 53 (1986), no. 1, 69–80. MR 861898, DOI 10.1007/BF02772670
Jairo Z. Gonçalves and Donald S. Passman, Free groups in normal subgroups of the multiplicative group of a division ring, J. Algebra 440 (2015), 128–144. MR 3373391, DOI 10.1016/j.jalgebra.2015.05.020
Jairo Z. Gonçalves and Mazi Shirvani, Algebraic elements as free factors in simple Artinian rings, Groups, rings and group rings, Contemp. Math., vol. 499, Amer. Math. Soc., Providence, RI, 2009, pp. 121–125. MR 2581930, DOI 10.1090/conm/499/09795
Bui Xuan Hai, Trinh Thanh Deo, and Mai Hoang Bien, On subgroups in division rings of type 2, Studia Sci. Math. Hungar. 49 (2012), no. 4, 549–557. MR 3098299, DOI 10.1556/SScMath.49.2012.4.1224
Bui Xuan Hai and Nguyen Kim Ngoc, A note on the existence of non-cyclic free subgroups in division rings, Arch. Math. (Basel) 101 (2013), no. 5, 437–443. MR 3125560, DOI 10.1007/s00013-013-0576-2
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.
B. Hartley, Free groups in normal subgroups of unit groups and arithmetic groups, Representation theory, group rings, and coding theory, Contemp. Math., vol. 93, Amer. Math. Soc., Providence, RI, 1989, pp. 173–177. MR 1003352, DOI 10.1090/conm/093/1003352
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, DOI 10.1016/j.jalgebra.2009.01.009
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, DOI 10.3318/pria.2014.114.04
T. Y. Lam, A first course in noncommutative rings, Graduate Texts in Mathematics, vol. 131, Springer-Verlag, New York, 1991. MR 1125071, DOI 10.1007/978-1-4684-0406-7
C. Bardos and M. Merigot, Asymptotic decay of the solution of a second-order elliptic equation in an unbounded domain. Applications to the spectral properties of a Hamiltonian, Proc. Roy. Soc. Edinburgh Sect. A 76 (1977), no. 4, 323–344. MR 477432, DOI 10.1017/S0308210500019673
A. I. Lichtman, Free subgroups of normal subgroups of the multiplicative group of skew fields, Proc. Amer. Math. Soc. 71 (1978), no. 2, 174–178. MR 480623, DOI 10.1090/S0002-9939-1978-0480623-2
A. I. Lichtman, Free subgroups in linear groups over some skew fields, J. Algebra 105 (1987), no. 1, 1–28. MR 871744, DOI 10.1016/0021-8693(87)90177-3
M. Mahdavi-Hezavehi, M. G. Mahmudi, and S. Yasamin, Finitely generated subnormal subgroups of $\textrm {GL}_n(D)$ are central, J. Algebra 225 (2000), no. 2, 517–521. MR 1741550, DOI 10.1006/jabr.1999.8096
J. Tits, Free subgroups in linear groups, J. Algebra 20 (1972), 250–270. MR 286898, DOI 10.1016/0021-8693(72)90058-0
G. M. Tomanov, Generalized group identities in linear groups, Mat. Sb. (N.S.) 123(165) (1984), no. 1, 35–49 (Russian). MR 728928
Zinovy Reichstein and Nikolaus Vonessen, Free subgroups of division algebras, Comm. Algebra 23 (1995), no. 6, 2181–2185. MR 1327133, DOI 10.1080/00927879508825339
Louis Halle Rowen, Polynomial identities in ring theory, Pure and Applied Mathematics, vol. 84, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London, 1980. MR 576061
M. Shirvani and B. A. F. Wehrfritz, Skew linear groups, London Mathematical Society Lecture Note Series, vol. 118, Cambridge University Press, Cambridge, 1986. MR 883801