Generalized power central group identities in almost subnormal subgroups of 𝐺𝐿_{𝑛}(𝐷)

Hai, B.; Khanh, H.; Bien, M.

doi:10.1090/spmj/1621

Generalized power central group identities in almost subnormal subgroups of $\operatorname {GL}_n(D)$
HTML articles powered by AMS MathViewer

by B. X. Hai, H. V. Khanh and M. H. Bien

St. Petersburg Math. J. 31 (2020), 739-749

DOI: https://doi.org/10.1090/spmj/1621

Published electronically: June 11, 2020

PDF | Request permission

Abstract:

This paper is devoted to almost subnormal subgroups of the general linear group $\mathrm {GL}_n(D)$ of degree $n\ge 1$ over a division ring $D$ that satisfy a generalized power central group identity.

References

S. Akbari and M. Arian-Nejad, Left Artinian algebraic algebras, Algebra Colloq. 8 (2001), no. 4, 463–470. MR 1865126
Mai Hoang Bien, Subnormal subgroups in division rings with generalized power central group identities, Arch. Math. (Basel) 106 (2016), no. 4, 315–321. MR 3475623, DOI 10.1007/s00013-016-0886-2
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
Katsuo Chiba, Skew fields with a nontrivial generalised power central rational identity, Bull. Austral. Math. Soc. 49 (1994), no. 1, 85–90. MR 1262677, DOI 10.1017/S0004972700016117
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
P. M. Cohn, Free rings and their relations, London Mathematical Society Monographs, No. 2, Academic Press, London-New York, 1971. MR 0371938
Trinh Thanh Deo, Mai Hoang Bien, and Bui Xuan Hai, On weakly locally finite division rings, Acta Math. Vietnam. 44 (2019), no. 2, 553–569. MR 3947983, DOI 10.1007/s40306-018-0292-x
Trinh Thanh Deo, Mai Hoang Bien, and Bui Xuan Hai, On division subrings normalized by almost subnormal subgroups in division rings, Period. Math. Hungar. 80 (2020), no. 1, 15–27. MR 4059860, DOI 10.1007/s10998-019-00282-5
Truong Huu Dung, On almost subnormal subgroups and maximal subgroup in skew linear groups, Int. Electron. J. Algebra 25 (2019), 35–42. MR 3904856, DOI 10.24330/ieja.504105
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 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
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
Bui Xuan Hai and Huynh Viet Khanh, Free subgroups in maximal subgroups of skew linear groups, Internat. J. Algebra Comput. 29 (2019), no. 3, 603–614. MR 3955824, DOI 10.1142/S0218196719500164
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
Israel N. Herstein, Claudio Procesi, and Murray Schacher, Algebraic valued functions on noncommutative rings, J. Algebra 36 (1975), no. 1, 128–150. MR 374185, DOI 10.1016/0021-8693(75)90160-X
I. N. Herstein, Multiplicative commutators in division rings, Israel J. Math. 31 (1978), no. 2, 180–188. MR 516254, DOI 10.1007/BF02760549
A. Kurosch, Ringtheoretische Probleme, die mit dem Burnsideschen Problem über periodische Gruppen in Zusammenhang stehen, Bull. Acad. Sci. URSS. Sér. Math. [Izvestia Akad. Nauk SSSR] 5 (1941), 233–240 (Russian, with German summary). MR 0005723
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
Leonid Makar-Limanov and Peter Malcolmson, Words periodic over the center of a division ring, Proc. Amer. Math. Soc. 93 (1985), no. 4, 590–592. MR 776184, DOI 10.1090/S0002-9939-1985-0776184-4
N. K. Ngoc, M. H. Bien, and B. X. Hai, Free subgroups in almost subnormal subgroups of general skew linear groups, Algebra i Analiz 28 (2016), no. 5, 220–235; English transl., St. Petersburg Math. J. 28 (2017), no. 5, 707–717. MR 3637590, DOI 10.1090/spmj/1468
Zinovy Reichstein and Nikolaus Vonessen, Free subgroups of division algebras, Comm. Algebra 23 (1995), no. 6, 2181–2185. MR 1327133, DOI 10.1080/00927879508825339
Joseph J. Rotman, Advanced modern algebra. Part 2, 3rd ed., Graduate Studies in Mathematics, vol. 180, American Mathematical Society, Providence, RI, 2017. With a foreword by Bruce Reznick. MR 3677125, DOI 10.1090/gsm/180
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
W. W. Stothers, Free subgroups of the free product of cyclic groups, Math. Comp. 32 (1978), no. 144, 1274–1280. MR 502015, DOI 10.1090/S0025-5718-1978-0502015-8
G. M. Tomanov, Generalized group identities in linear groups, Mat. Sb. (N.S.) 123(165) (1984), no. 1, 35–49 (Russian). MR 728928