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

Question

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

Answer 1

Consider an irreducible linear groupG ⊆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(_kGV ), 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-algebraF 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.