Maximal subgroups of subnormal subgroups of \(\text{GL}_n(D)\) with finite conjugacy classes. (English) Zbl 1222.20035
Let \(D\) be a division ring. It is very well-known that if \(D^*\) is an FC-group, then \(D\) is a field. In the article under review the authors consider maximal subgroups of a subnormal subgroup \(N\) of \(\text{GL}_n(D)\) for \(n\geq 1\). The main result states that if a maximal subgroup \(M\) of \(N\) is an FC-group, then \(M\) is contained in the multiplicative group of some subfield of \(M_n(D)\).
Reviewer: Bui Xuan Hai (Ho Chi Minh City)
MSC:
| 20H25 | Other matrix groups over rings |
| 20E15 | Chains and lattices of subgroups, subnormal subgroups |
| 20E28 | Maximal subgroups |
| 20F24 | FC-groups and their generalizations |
| 16U60 | Units, groups of units (associative rings and algebras) |
| 16K40 | Infinite-dimensional and general division rings |
| 12E15 | Skew fields, division rings |
| 15B33 | Matrices over special rings (quaternions, finite fields, etc.) |
Keywords:
division rings; FC-groups; maximal subgroups; subnormal subgroups; multiplicative groups; maximal subfieldsReferences:
| [1] | DOI: 10.1016/S0021-8693(02)00549-5 · Zbl 1016.20031 · doi:10.1016/S0021-8693(02)00549-5 |
| [2] | DOI: 10.1090/S0002-9947-1955-0074393-9 · doi:10.1090/S0002-9947-1955-0074393-9 |
| [3] | Dixon J. D., The Structure of Linear Groups (1971) · Zbl 0232.20079 |
| [4] | DOI: 10.1017/CBO9780511661907 · doi:10.1017/CBO9780511661907 |
| [5] | DOI: 10.1016/j.jalgebra.2004.02.018 · Zbl 1073.20041 · doi:10.1016/j.jalgebra.2004.02.018 |
| [6] | DOI: 10.1007/BF02760549 · Zbl 0394.16015 · doi:10.1007/BF02760549 |
| [7] | DOI: 10.1007/s00229-009-0290-3 · Zbl 1184.20043 · doi:10.1007/s00229-009-0290-3 |
| [8] | DOI: 10.1007/s00229-007-0124-0 · Zbl 1132.20030 · doi:10.1007/s00229-007-0124-0 |
| [9] | Kiani D., Algebra Colloq. 12 pp 461– (2005) |
| [10] | DOI: 10.1007/978-1-4419-8616-0 · doi:10.1007/978-1-4419-8616-0 |
| [11] | Mahdavi-Hezavehi M., Algebra colloq. 5 pp 361– (1998) |
| [12] | DOI: 10.1006/jabr.1999.8096 · Zbl 0947.20032 · doi:10.1006/jabr.1999.8096 |
| [13] | DOI: 10.1006/jabr.2001.8782 · Zbl 0984.16029 · doi:10.1006/jabr.2001.8782 |
| [14] | Passman D. S., The algebraic structure of group rings (1977) · Zbl 0368.16003 |
| [15] | Rowen L. H., Polynomial Identities in Ring Theory (1980) · Zbl 0461.16001 |
| [16] | Scott W. R., Group Theory (1987) |
| [17] | Shirvani M., Skew Linear Groups (1986) · Zbl 0602.20046 |
| [18] | Wehrfritz A. F., Infinite Linear Groups (1973) · Zbl 0261.20038 |
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