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Absolute irreducibility for finitary linear groups. (English) Zbl 0827.20057
In this short note the author contributes to the present rapid development of the theory of finitary linear groups by considering absolute irreducibility. Let \(V\) be a vector space over the field \(K\) and \(L\) an extension field of \(K\). Then \(\text{GL}(V)\) acts in the obvious way on \(\text{GL}(V^L)\) for \(V^L=L\otimes_KV\). Call a subgroup \(G\) of \(\text{GL}(V)\) absolutely irreducible if \(V^L\) is irreducible as \(LG\)-module for every extension field \(L\) of \(K\). The author proves the following, thus extending the finite-dimensional (linear) case.
Theorem. Let \(V\) be a vector space over the field \(K\) and let \(G\) be an irreducible subgroup of \(\text{FGL}(V)\). Then there exists a finite field extension \(L/K\) such that \(V\) is also a vector space over \(L\), and such that \(V\) is absolutely irreducible as \(LG\)-module. Here the degree of the field extension \(L/K\) is finite and divides the \(K\)-dimension of every finite-dimensional \(L\)-subspace of \(V\). Examples of \(L\)-subspaces are \(C_V(F)\) and \([V,F]\) for every finitely generated subgroup \(F\) of \(G\).
Slightly surprising perhaps is the finiteness of the degree \(L/K\). In the finite-dimensional case, of course, this is obviously finite.

MSC:
20H20 Other matrix groups over fields
20G05 Representation theory for linear algebraic groups
20C07 Group rings of infinite groups and their modules (group-theoretic aspects)
20E07 Subgroup theorems; subgroup growth
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References:
[1] I.N. Herstein , Noncommutative Rings , J. Wiley & Sons ( 1971 ). MR 227205 | Zbl 0177.05801 · Zbl 0177.05801
[2] B.A.F. Wehrfritz , Infinite Linear Groups , Springer-Verlag , Berlin - Heidelberg - New York ( 1973 ). MR 335656 | Zbl 0261.20038 · Zbl 0261.20038
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