**Definition**. A group is called **hopfian** if every surjective homomorphism is an isomorphism. Clearly every finite group is hopfian.

**Problem**. Prove that is not hopfian.

**Solution**. Define by and and extend it to homomorphically. Note that is well-defined because

We also have and so is surjective. To prove that is not an isomorphism, let Then but

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