Locally solvable and solvable-by-finite maximal subgroups of GLn(D)
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Huynh Viet Khanh
Vietnam National University, Faculty of Mathematics and Computer Science
Bui Xuan Hai
Vietnam National University, Faculty of Mathematics and Computer Science
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.
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Khanh, Huynh Viet; Hai, Bui Xuan. «Locally solvable and solvable-by-finite maximal subgroups of GLn(D)». Publicacions Matemàtiques, 2022, vol.VOL 66, núm. 1, p. 77-97, http://raco.cat/index.php/PublicacionsMatematiques/article/view/396410.
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Preprint 2019. arXiv:1908.04925.
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Algebra Colloq. 5(4) (1998), 361–370.
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subnormal subgroups of GLn(D) are central, J. Algebra 225(2) (2000), 517–521.
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N. K. Ngoc, M. H. Bien, and B. X. Hai, Free subgroups in almost subnormal
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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. DOI: 10.1016/
j.jalgebra.2013.09.042.
L. H. Rowen, “Polynomial Identities in Ring Theory”, Pure and Applied Mathematics 84, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New
York-London, 1980. DOI: 10.1016/s0079-8169(08)x6032-5.
M. Shirvani and B. A. F. Wehrfritz, “Skew Linear Groups”, London Mathematical Society Lecture Note Series 118, Cambridge University Press, Cambridge, 1986. DOI: 10.1017/CBO9780511600630.
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 und ihrer Grenzgebiete 76, Springer-Verlag, New York-Heidelberg, 1973.
DOI: 10.1007/978-3-642-87081-1.
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/
40.4.501.
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.
DOI: 10.1016/S0021-8693(02)00549-5.
T. T Deo, M. H. Bien, and B. X. Hai, On division subrings normalized by
almost subnormal subgroups in division rings, Period. Math. Hungar. 80(1)
(2020), 15–27. DOI: 10.1007/s10998-019-00282-5.
H. R. Dorbidi, R. Fallah-Moghaddam, and M. Mahdavi-Hezavehi, Soluble maximal subgroups in GLn(D), J. Algebra Appl. 10(6) (2011), 1371–1382.
DOI: 10.1142/S0219498811005233.
P. K. Draxl, “Skew Fields”, London Mathematical Society Lecture Note Series 81, Cambridge University Press, Cambridge, 1983. DOI: 10.1017/CBO978051
1661907.
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 Student Texts 61, Cambridge University Press, Cambridge, 2004. DOI: 10.1017/
CBO9780511841699.
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 2020.
arXiv:2002.06522.
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. DOI: 10.
1142/S0218196719500164.
B. Hartley, Free groups in normal subgroups of unit groups and arithmetic groups, in: “Representation Theory, Group Rings, and Coding Theory”,
Contemp. Math. 93, Amer. Math. Soc., Providence, RI, 1989, pp. 173–177.
DOI: 10.1090/conm/093/1003352.
R. Hazrat, M. Mahdavi-Hezavehi, and M. Motiee, Multiplicative groups of
division rings, Math. Proc. R. Ir. Acad. 114A(1) (2014), 37–114. DOI: 10.3318/
pria.2014.114.04.
R. Hazrat and A. R. Wadsworth, On maximal subgroups of the multiplicative
group of a division algebra, J. Algebra 322(7) (2009), 2528–2543. DOI: 10.1016/
j.jalgebra.2009.01.009
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: 10.1007/
978-1-4419-8616-0.
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), 517–521.
DOI: 10.1006/jabr.1999.8096.
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) (2016), 220–235;
reprinted in St. Petersburg Math. J. 28(5) (2017), 707–717. DOI: 10.1090/spmj/
1468.
D. S. Passman, “The Algebraic Structure of Group Rings”, Pure and Applied Mathematics, Wiley-Interscience [John Wiley & Sons], New York-LondonSydney, 1977.
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. DOI: 10.1016/
j.jalgebra.2013.09.042.
L. H. Rowen, “Polynomial Identities in Ring Theory”, Pure and Applied Mathematics 84, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New
York-London, 1980. DOI: 10.1016/s0079-8169(08)x6032-5.
M. Shirvani and B. A. F. Wehrfritz, “Skew Linear Groups”, London Mathematical Society Lecture Note Series 118, Cambridge University Press, Cambridge, 1986. DOI: 10.1017/CBO9780511600630.
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 und ihrer Grenzgebiete 76, Springer-Verlag, New York-Heidelberg, 1973.
DOI: 10.1007/978-3-642-87081-1.
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/
40.4.501.
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.