Is this proof correct? Give a group theoretic proof that Q under addition is not isomorphic to R^+ under multiplication.

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I am self-studying abstract algebra so sometimes having a proof critiqued really helps me out. I'm not looking for the correct proof, just criticism of my proof in terms of logic or writing style. I'm a former math major (graduated) and I may apply to graduate school someday. Thanks for any help.

My write-up is below.

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