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

Table of contents
- Solution iv

Solution

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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