Absolute irreducibility for finitary linear groups.

*(English)*Zbl 0827.20057In 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.

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.

Reviewer: B.A.F.Wehrfritz (London)

##### 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 |

##### Keywords:

absolutely irreducible modules; finitary linear groups; absolute irreducibility; irreducible subgroups; finite field extensions; finitely generated subgroups
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\textit{F. Leinen}, Rend. Semin. Mat. Univ. Padova 92, 59--61 (1994; Zbl 0827.20057)

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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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