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Irreducible representations of periodic finitary linear groups. (English) Zbl 0856.20036

Let \(K\) be a field and \(V\) a vector space of infinite dimension over \(K\). The elements \(g\in\text{Aut}_K(V)\) such that \(\dim(g-\text{Id})V<\infty\) form a group \(\text{FGL}(V)\) called finitary linear. Let \(K_0\) denote the algebraic closure of the prime subfield of \(K\) in \(K\). Let \(G\subset\text{FGL}(V)\) be a periodic subgroup. The main theorem says that there is a basis \(B\) of \(V\) such that the matrices of \(G\) with respect to \(B\) have their entries in \(K_0\).

MSC:

20H20 Other matrix groups over fields
20G05 Representation theory for linear algebraic groups
20G15 Linear algebraic groups over arbitrary fields
20F50 Periodic groups; locally finite groups
20E07 Subgroup theorems; subgroup growth
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