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# Homomorphism and First Isomorphism Theorem

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Let R>0 be the group of positive real numbers under multiplication. Let CX be the group of nonzero complex numbers under mu!tiplication. Let S1 = {a + bi such that a^2 + b^2 = 1) be the subgroup of C consisting of all complex numbers of absolute value 1. Note that is normal in Cx since Cx is abelian. Prove that CX/S1 is isomorphic to R>0. [Hint: Find a homomorphism from Cx to R>0 and use the First Isomorphism Theorem.]

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