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ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2021/ia/paper-3/8d/a/i/solution

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Past exam of the mathematics course of the University of Cambridge / 2021 / ia / Paper 3 / 8D / a / i / Solution by

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

Identify

A

with

{(a, 1) : a \in A}

. Componentwise use of the stated multiplication shows that this set contains the identity and is closed under products and inverses, so it is a subgroup. It has the two cosets

A, A (e, - 1),

(1)

and therefore has index two. Every index-two subgroup is normal, so

A ◃ D (A) .

(2)

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