Recall that R* is the set of nonzero real numbers, which forms a group under multiplication. (1) Prove that N = {(x, )|x R*} is a subgroup of R* x R*. (2) For (a, b), (a', b') e R* x R*, give necessary and sufficient conditions for N(a, b) = N(a', b'). (3) Define a surjective group homomorphism f: R* x R* R* whose kernel is N. Use the First Isomorphism Theorem to determine the structure of R*/N.

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