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Subgroups containing a torus and related to the quotient field of a factorial ring. (Russian) Zbl 1081.20055
Let \(R\) be a factorial integral ring with identity and let \(k\) be the quotient field of \(R\). Let \(d\) be a square-free element of \(R\) and let \(K\) be a splitting field of \(x^n-d\). It is well known that \(K^*\) can be isomorphically embedded into \(\operatorname{Aut}_k(K)\simeq\text{GL}_n(k)\). The authors call the image \(T(d)\) of \(K^*\) in \(\text{GL}_n(k)\) a nonsplit maximal torus. They describe subgroups of \(\text{GL}_n(k)\) maximal among those containing \(T(d)\) and not contained in \(\text{SL}_n(k)\).
MSC:
20G15 Linear algebraic groups over arbitrary fields
20E15 Chains and lattices of subgroups, subnormal subgroups
20E07 Subgroup theorems; subgroup growth
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