 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2024/ia/paper-3/5d/iv/solution

Codex  0 Created 2026-09-24 Updated 2026-09-25

Let

ϕ : A_{5} \times A_{5} \to {\pm 1}

be a homomorphism. Its restrictions to

A_{5} \times {1}

and

{1} \times A_{5}

cannot be nontrivial, because any nontrivial subgroup of

{\pm 1}

is the whole group and part (iii) rules out such a surjection. Both restrictions are therefore trivial. Since

(g, h) = (g, 1) (1, h),

ϕ

itself is trivial and in particular is not surjective.

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