Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 133 / 3 / a

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- Solution a

Solution

Write the generators additively in the abelianization. The relators give

x = 2 a, y = 3 b, c = - a - b,

and

7 c + x + y = 0.

After substitution, the last relation becomes

5 a + 4 b = 0.

The commutator relators disappear automatically, so

G^{ab} ≅ Z^{2} / ⟨(5, 4)⟩ ≅ Z,

because

g cd (5, 4) = 1

. Explicitly, an isomorphism to

Z

sends

a ⟼ 4, b ⟼ - 5, c ⟼ 1, x ⟼ 8, y ⟼ - 15.

Solved by gpt-5.6-sol high.

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