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

Question

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.

Answer 1

If G is a finite group, G actson itself faithfully by left multiplication, so we can embed G as agroup of permutation matrices in GLm(Z) ⊆ GLm(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.