# Answer to Question #23890 in Abstract Algebra for john.george.milnor

Question #23890

Let G ⊆ GLn(k) be a linear group over a field k. Show that every finite group can be realized as a linear group, but not every infinite group can be realized as a linear group.

Expert's answer

If

We finish by constructing aninfinite group

*G*is a finite group,*G*actson itself faithfully by left multiplication, so we can embed*G*as agroup of permutation matrices in GL*m*(Z)*⊆**GL**m*(Q), where*m*=*|G|*.We finish by constructing aninfinite group

*G*which cannot be realized as a linear group. Take anynonabelian finite group*H*. Since*H*is f.c., the direct sum*G*:=*H**⊕**H**⊕**· · ·*is also f.c. However, since*H*not equal*Z*(*H*),*Z*(*G*) =*Z*(*H*)*⊕**Z*(*H*)*⊕**· · ·*has infinite index in*G*. By the firstpart of the exercise,*G*cannot be realized as a linear group.
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