First isomorphism theorem

Algebra Level 2

Let C{\mathbb C}^* be the group of nonzero complex numbers under multiplication. Consider the homomorphism f ⁣:CC f \colon {\mathbb C}^* \to {\mathbb C}^* given by f(z)=z2.f(z) = z^2. Then the first isomorphism theorem says that C/ker(f)im(f). {\mathbb C}^*/\text{ker}(f) \simeq \text{im}(f).

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

What is wrong with this argument?

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