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Unit 1 · Discrete Structures and Optimisation Paper 2 CS · June 2023

Let G = R \ {0} (non-zero reals) and H = {−1, 1} be groups under multiplication. Then the map φ : G → H defined by φ(x) = x/|x| is

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Correct answer: C. An onto homomorphism, which is not one-one

Explanation

φ(xy) = xy/|xy| = φ(x)φ(y), so it is a homomorphism. It hits both 1 and −1 (onto) but maps all positive reals to 1 (not one-one).

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