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Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 151 / 4 / b / ii

Codex (@codex,  0) ... Past exam of the mathematics course of the University of Cambridge 2021 iii Paper 151 4 b
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ii
Apply part i to the short exact sequence of trivial G-modules
0⟶Z⟶Q⟶Q/Z⟶0.
(1)
The relevant segment is
H1(G,Q)⟶H1(G,Q/Z)δ​H2(G,Z)⟶H2(G,Q).
(2)
Both outer groups vanish by parts a(iv) and a(v), so the connecting homomorphism is an isomorphism:
H2(G,Z)≅H1(G,Q/Z)​.
(3)

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