# Answer to Question #23891 in Abstract Algebra for john.george.milnor

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(*), which is, by Schur’s Lemma, a (finite dimensional)division algebra over*_{kG}V*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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