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Soluble-by-periodic skew linear groups

Published online by Cambridge University Press:  24 October 2008

B. A. F. Wehrfritz
B. A. F. Wehrfritz
Affiliation:
Queen Mary College, London El 4NS
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Extract

Let D be a division ring with central subfield F, n a positive integer and G a subgroup of GL(n, D) such that the F-subalgebra F[G] generated by G is the full matrix algebra Dn×n. If G is soluble then Snider [9] proves that G is abelian by locally finite. He also shows that this locally finite image of G can be any locally finite group. Of course not every abelian by locally finite group is soluble. This suggests that Snider's conclusion should apply to some wider class of groups.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 96 , Issue 3 , November 1984 , pp. 379 - 389
Copyright
Copyright © Cambridge Philosophical Society 1984

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References

REFERENCES

[1]Carter, R. W.. Simple Groups of Lie Type (Wiley, 1972).Google Scholar
[2]Chatters, A. W. and Hajarnavis, C. R.. Rings with Chain Conditions (Pitman, 1980).Google Scholar
[3]Cohn, P. M.. Algebra, vol. 2 (Wiley, 1977).Google Scholar
[4]Formanek, E. and Jategaonkar, A. V.. Subrings of Noetherian rings. Proc. Amer. Math. Soc. 46 (1974), 181–186.CrossRefGoogle Scholar
[5]Hartley, B. and Shahabi Shojaei, M. A.Finite groups of matrices over division rings. Math. Proc. Cambridge Philos. Soc. 92 (1982), 55–64.CrossRefGoogle Scholar
[6]Kegel, O. H. and Wehrfritz, B. A. F.. Locally Finite Groups (North-Holland, 1973).Google Scholar
[7]Passman, D. S.. The Algebraic Structure of Group Rings (Wiley, 1977).Google Scholar
[8]Robinson, D. J. S.. Finiteness Conditions and Generalized Soluble Groups, vol. 1 (Springer-Verlag, 1972).Google Scholar
[9]Snider, R. L.. Solvable linear groups over division rings. (Preprint 1983.)CrossRefGoogle Scholar
[10]Steinberg, R.. Lectures on Chevalley Groups. Yale University Lecture Notes (1967).Google Scholar
[11]Thomas, S.. The classification of the simple periodic linear groups. Arch. Math. (Basel), 41 (1983), 103–116.CrossRefGoogle Scholar
[12]Wehrfritz, B. A. F.. Infinite Linear Groups (Springer-Verlag, 1973).CrossRefGoogle Scholar
[13]Wehrfritz, B. F.. The rank of a linear ρ-group; an apology. J. London Math. Soc. (2), 21 (1980), 237–243.CrossRefGoogle Scholar
[14]Zalesskiῐ, A.. The structure of several classes of matrix groups over a division ring (Russian). Sibirsk. Mat. Ž. 8 (1967), 1284–1289 [= Siberian Math. J. 8 (1967), 978–988].Google Scholar