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