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:
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