G

0 \to H^{0} (G, M_{1}) \to H^{0} (G, M_{2}) \to H^{0} (G, M_{3}) δ H^{1} (G, M_{1}) \to H^{1} (G, M_{2}) \to \dots .

ii

Solution

Apply part i to the short exact sequence of trivial

G

-modules

0 ⟶ Z ⟶ Q ⟶ Q / Z ⟶ 0.

The relevant segment is

H^{1} (G, Q) ⟶ H^{1} (G, Q / Z) δ H^{2} (G, Z) ⟶ H^{2} (G, Q) .

Both outer groups vanish by parts a(iv) and a(v), so the connecting homomorphism is an isomorphism:

H^{2} (G, Z) ≅ H^{1} (G, Q / Z) .

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