First isomorphism theorem

Algebra Level 5

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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