 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2026/iii/paper-133/1/a/ii/solution

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

For

ϕ (a) = 2, ϕ (b) = 3,

ϕ : F_{2} \to Z

is surjective because

g cd (2, 3) = 1

. Again identify the cosets of

ker ϕ

with

Z

v_{n}

v_{n} a v_{n + 2}, v_{n} b v_{n + 3} .

This labelled graph is connected: integer combinations of

2

and

3

reach every vertex. Each vertex has one incoming and one outgoing edge of each label, as required for a covering graph of the two-petalled rose.

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