First isomorphism theorem

Algebra Level 2

Let ${\mathbb C}^*$ be the group of nonzero complex numbers under multiplication. Consider the homomorphism $f \colon {\mathbb C}^* \to {\mathbb C}^*$ given by $f(z) = z^2.$ Then the first isomorphism theorem says that ${\mathbb C}^*/\text{ker}(f) \simeq \text{im}(f).$

Now $\text{ker}(f) = \{ \pm 1\},$ and $\text{im}(f) = {\mathbb C}^*$ (that is, $f$ is onto), so the conclusion is that ${\mathbb C}^*/\{\pm 1\} \simeq {\mathbb C}^*,$ i.e. ${\mathbb C}^*$ is isomorphic to a nontrivial quotient of itself.

What is wrong with this argument?

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