Abstract
The aim of this note is to investigate the structure of skew linear groups of finite rank. Among our results, it is proved that a subgroup G of \(\mathrm {GL}_n(D)\) has finite rank if and only if there exists a solvable normal subgroup N in G of finite rank such that the factor group G/N is finite provided D is a locally finite division ring which is not necessarily commutative.
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References
Brandl, R., Franciosi, S., de Giovanni, F.: On the Wielandt subgroup of infinite soluble groups. Glasgow Math. J. 32, 121–125 (1990)
Čarin, V.S.: On locally solvable groups of finite rank. Mat. Sb. N.S. 41(83), 37–48 (1957)
Chernikov, N.S.: A theorem on groups of finite special rank. Ukrain. Mat. Zh. 42(7), 962–970 (1990); translation in Ukrainian Math. J. 42(7), 855–861 (1990)
Detinko, A.S., Flannery, D.L., O‘Brien, E.A.: Algorithms for linear groups of finite rank. J. Algebra 393, 187–196 (2013)
Dixon, M.R., Kurdachenko, L.A., Polyakov, N.V.: On some ranks of infinite groups. Ric. Math. 56, 43–59 (2007)
Dixon, M.R., Evan, M.J., Smith, H.: Locally (solvable-by-finite) groups of finite rank. J. Algebra 182, 756–769 (1996)
Dixon, M.R.: Sylow Theory, Formations and Fitting Classes in Locally Finite Groups. Series in Algebra, 2. World Scientific Publishing Co., Inc., River Edge, NJ (1994)
Draxl, P.K.: Skew Fields. London Mathematical Society Lecture Note Series, 81. Cambridge University Press, Cambridge (1983)
Kargapolov, M.I.: On soluble groups of finite rank. Algebra i Logika 1, 37–44 (1962)
Hai, B.X., Khanh, H.V.: Multiplicative subgroups in weakly locally finite division rings. Acta Math. Vietnam 46, 779–794 (2021)
Khanh, H.V., Hai, B.X.: Locally solvable and solvable-by-finite maximal subgroups of \({\rm GL}_n (D)\). Publ. Mat. 66(1), 77–97 (2022)
Hartley, B.: Free groups in normal subgroups of unit groups and arithmetic groups. Contemp. Math. 93, 173–177 (1989)
Lubotzky, A., Mann, A.: Residually finite groups of finite rank. Math. Proc. Cambridge Philos. Soc. 106, 385–388 (1989)
Mahdavi-Hezavehi, M., Mahmudi, M.G., Yasamin, S.: Finitely generated subnormal subgroups of \({\rm GL}_n (D)\). J. Algebra 225, 517–521 (2000)
Mal‘cev, A.I., On isomorphic matrix representations of infinite groups. Rec. Math. [Mat. Sbornik] N.S. 8(50), 405–422 (1940)
Mal’cev, A.I.: On groups of finite rank. Mat. Sbornik N.S. 22, 350–352 (1948)
Merzlyakov, Yu.I.: On locally soluble groups of finite rank. Algebra i Logika 3, 5–16 (1964)
Ngoc, N.K., Bien, M.H., Hai, B.X.: Free subgroups in almost subnormal subgroups of general skew linear groups. Algebra i Analiz 28(5), 220–235 (2016); reprinted in St. Petersburg Math. J. 28(5), 707–717 (2017)
Platonov, V.P.: On a problem of Mal‘cev. Mat. Sbornik N.S. 79, 621–624 (1969)
Robinson, Derek J.S.: A Course in the Theory of Groups. Second Edition. Graduate Texts in Mathematics, 80. Springer-Verlag, New York (1996)
Shirvani, M.: On soluble-by-finite subgroup of division algebras. J. Algebra 294, 255–277 (2005)
Shirvani, M., Wehrfritz, B.A.F.: Skew Linear Groups. Cambridge University Press, Cambridge (1986)
Stuth, C.J.: A generalization of the Cartan-Brauer-Hua theorem. Proc. Amer. Math. Soc. 15(2), 211–217 (1964)
Shunkov, V.P.: On locally finite groups of finite rank. Algebra Logic 10, 127–142 (1971)
Wehrfritz, B.A.F.: Infinite Linear Groups. Springer, Berlin (1973)
Wehrfritz, B.A.F.: On groups of finite rank. Publ. Mat. 65, 599–613 (2021)
Acknowledgements
The authors express their sincere gratitude to the anonymous referee for his/her agreement for the presentation of his/her additional proof of Theorem 6.
Funding
This research is funded by Vietnam National University HoChiMinh City (VNUHCM) under grant number T2022-18-03.
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Van Chua, L., Bien, M.H. & Hai, B.X. A note on skew linear groups of finite rank. Arch. Math. 119, 113–120 (2022). https://doi.org/10.1007/s00013-022-01732-2
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DOI: https://doi.org/10.1007/s00013-022-01732-2