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morphism

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Morphism is sort of on the right lines. The property that you're looking for is called divisibility. Q has it and Z doesn't. The same idea also shows that Q\{0} under multiplication is not isomorphic to R\{0} under multiplication. (One can provide an elementary counter argument based purely on set theory, of course.)

Alternatively, Z is cyclic. Can you prove that Q isn't? Actually it isn't that alternative, really. Try thinking about a map g from Z to Q. Any group hom is determined completely by where it sends 1 in Z. Can g(1)/2 be in the image?

Alternatively, Z is cyclic. Can you prove that Q isn't? Actually it isn't that alternative, really. Try thinking about a map g from Z to Q. Any group hom is determined completely by where it sends 1 in Z. Can g(1)/2 be in the image?

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