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

1. ANot a homomorphism
2. BA one-one homomorphism, which is not onto
3. CAn onto homomorphism, which is not one to one
4. DAn homomorphism

Pick an option to check yourself, or open the answer below.

Show answer & explanation

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

Explanation

φ(x) is the sign of x, and sign(xy) = sign(x)·sign(y), so φ is a homomorphism. It hits both 1 and −1 (onto) but sends every positive number to 1 (not one-to-one).