Solution
= Solution
Let $\phi:A_5\times A_5\to\{\pm1\}$ be a homomorphism. Its restrictions to $A_5\times\{1\}$ and $\{1\}\times A_5$ cannot be nontrivial, because any nontrivial <subgroup> of $\{\pm1\}$ is the whole <group> and part (iii) rules out such a surjection. Both restrictions are therefore trivial. Since
$$
(g,h)=(g,1)(1,h),
$$
$\phi$ itself is trivial and in particular is not surjective.