Solution
= Solution
If a homomorphism $A_5\to\{\pm1\}$ 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.
= Solution
If a homomorphism $A_5\to\{\pm1\}$ 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.