Locally solvable and solvable-by-finite maximal subgroups of GLn(D)

• Autores: Huynh Viet Khanh, Bui Xuan Hai
• Localización: Publicacions matematiques, ISSN 0214-1493, Vol. 66, Nº 1, 2022, págs. 77-97
• Idioma: inglés
• Enlaces
  • Texto completo
• Resumen

  • This paper aims to study solvable-by-finite and locally solvable maximal subgroups of an almost subnormal subgroup of the general skew linear group GLn(D) over a division ring D. It turns out that in the case where D is non-commutative, if such maximal subgroups exist, then either it is abelian or [D : F] < ∞. Also, if F is an infinite field and n ≥ 5, then every locally solvable maximal subgroup of a normal subgroup of GLn(F) is abelian.

• Referencias bibliográficas
  • S. Akbari, R. Ebrahimian, H. Momenaee Kermani, and A. Salehi Golsefidy, Maximal subgroups of GLn(D), J. Algebra 259(1) (2003), 201–225. DOI:...
  • T. T Deo, M. H. Bien, and B. X. Hai, On division subrings normalized by almost subnormal subgroups in division rings, Period. Math. Hungar....
  • H. R. Dorbidi, R. Fallah-Moghaddam, and M. Mahdavi-Hezavehi, Soluble maximal subgroups in GLn(D), J. Algebra Appl. 10(6) (2011), 1371–1382....
  • P. K. Draxl, “Skew Fields”, London Mathematical Society Lecture Note Series 81, Cambridge University Press, Cambridge, 1983. DOI: 10.1017/CBO978051...
  • R. Ebrahimian, Nilpotent maximal subgroups of GLn(D), J. Algebra 280(1) (2004), 244–248. DOI: 10.1016/j.jalgebra.2004.02.018.
  • R. Fallah-Moghaddam, Maximal subgroups of SLn(D), J. Algebra 531 (2019), 70–82. DOI: 10.1016/j.jalgebra.2019.05.003.
  • K. R. Goodearl and R. B. Warfield, Jr., “An Introduction to Noncommutative Noetherian Rings”, Second edition, London Mathematical Society...
  • B. X. Hai, B.-M. Bui-Xuan, L. V. Chua, and M. H. Bien, Intersection graphs of almost subnormal subgroups in general skew linear groups, Preprint...
  • B. H. Hai and H. V. Khanh, Free subgroups in maximal subgroups of skew linear groups, Internat. J. Algebra Comput. 29(3) (2019), 603–614....
  • B. Hartley, Free groups in normal subgroups of unit groups and arithmetic groups, in: “Representation Theory, Group Rings, and Coding Theory”,...
  • R. Hazrat, M. Mahdavi-Hezavehi, and M. Motiee, Multiplicative groups of division rings, Math. Proc. R. Ir. Acad. 114A(1) (2014), 37–114. DOI:...
  • R. Hazrat and A. R. Wadsworth, On maximal subgroups of the multiplicative group of a division algebra, J. Algebra 322(7) (2009), 2528–2543....
  • I. N. Herstein, Multiplicative commutators in division rings, Israel J. Math. 31(2) (1978), 180–188. DOI: 10.1007/BF02760549.
  • H. V. Khanh and B. X. Hai, A note on solvable maximal subgroups in subnormal subgroups of GLn(D), Preprint 2018. arXiv:1809.00356.
  • H. V. Khanh and B. X. Hai, On almost subnormal subgroups in division rings, Preprint 2019. arXiv:1908.04925.
  • T. Y. Lam, “A First Course in Noncommutative Rings”, Second edition, Graduate Texts in Mathematics 131, Springer-Verlag, New York, 2001. DOI:...
  • C. Lanski, Solvable subgroups in prime rings, Proc. Amer. Math. Soc. 82(4) (1981), 533–537. DOI: 10.2307/2043766.
  • M. Mahdavi-Hezavehi and S. Akbari, Some special subgroups of GLn(D), Algebra Colloq. 5(4) (1998), 361–370.
  • M. Mahdavi-Hezavehi, M. G. Mahmudi, and S. Yasamin, Finitely generated subnormal subgroups of GLn(D) are central, J. Algebra 225(2) (2000),...
  • 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(5)...
  • D. S. Passman, “The Algebraic Structure of Group Rings”, Pure and Applied Mathematics, Wiley-Interscience [John Wiley & Sons], New York-LondonSydney,...
  • M. Ramezan-Nassab and D. Kiani, Some skew linear groups with Engel’s condition, J. Group Theory 15(4) (2012), 529–541. DOI: 10.1515/jgt-2012-0003.
  • M. Ramezan-Nassab and D. Kiani, Nilpotent and polycyclic-by-finite maximal subgroups of skew linear groups, J. Algebra 399 (2014), 269–276....
  • L. H. Rowen, “Polynomial Identities in Ring Theory”, Pure and Applied Mathematics 84, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers],...
  • M. Shirvani and B. A. F. Wehrfritz, “Skew Linear Groups”, London Mathematical Society Lecture Note Series 118, Cambridge University Press,...
  • C. J. Stuth, A generalization of the Cartan–Brauer–Hua theorem, Proc. Amer. Math. Soc. 15 (1964), 211–217. DOI: 10.2307/2034037.
  • B. A. F. Wehrfritz, “Infinite Linear Groups”, An Account of the Grouptheoretic Properties of Infinite Groups of Matrices, Ergebnisse der Matematik...
  • B. A. F. Wehrfritz, Soluble-by-periodic skew linear groups, Math. Proc. Cambridge Philos. Soc. 96(3) (1984), 379–389. DOI: 10.1017/S0305004100062307.
  • B. A. F. Wehrfritz, Goldie subrings of Artinian rings generated by groups, Quart. J. Math. Oxford Ser. (2) 40(4) (1989), 501–512. DOI: 10.1093/qmath/...
  • B. A. F. Wehrfritz, Crossed product criteria and skew linear groups II, Michigan Math. J. 37(2) (1990), 293–303. DOI: 10.1307/mmj/1029004136.
  • B. A. F. Wehrfritz, Crossed product criteria and skew linear groups, J. Algebra 141(2) (1991), 321–353. DOI: 10.1016/0021-8693(91)90235-Z.
  • B. A. F. Wehrfritz, Normalizers of nilpotent subgroups of division rings, Q. J. Math. 58(1) (2007), 127–135. DOI: 10.1093/qmath/hal012.