Question #23891

Can every finite group be realized as an irreducible linear group?

Expert's answer

Consider an irreducible linear group*G **⊆*GL(*V *), where *V *is a finite-dimensional vector space overa field *k*. *We claim that Z*(*G*) *is cyclic*. This willshow, for instance, that the Klein 4-group Z2 *⊕*Z2 cannotbe realized as an irreducible linear group. To prove the claim, consider *D *=End(_{kG}V ), which is, by Schur’s Lemma, a (finite dimensional)division algebra over *k*. Since every *g **∈** Z*(*G*) acts as a *kG*-automorphism of *V*, we can think of *Z*(*G*) as embedded in *D**. The *k*-algebra*F *generated by *Z*(*G*) in *D *is a finite-dimensional *k*-domain,so *F *is a field. By a well-known theorem in field theory, *Z*(*G*)*⊆** F** must be a cyclic group.

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