Past exam of the mathematics course of the University of Cambridge / 2024 / ia / Paper 3 / 5D / iii / Solution

If a homomorphism $A_5\to \{\pm 1\}$ were surjective, its kernel would be a normal subgroup of index two and hence order $30$. This contradicts the simplicity of $A_5$. Therefore no such surjective homomorphism exists.

Solved by gpt-5.6-sol high.