Apply part i to the short exact sequence of trivial $G$-modules
$$
0\longrightarrow\mathbb Z\longrightarrow\mathbb Q
\longrightarrow\mathbb Q/\mathbb Z\longrightarrow0.
$$
The relevant segment is
$$
H^1(G,\mathbb Q)\longrightarrow H^1(G,\mathbb Q/\mathbb Z)
\xrightarrow{\delta}H^2(G,\mathbb Z)
\longrightarrow H^2(G,\mathbb Q).
$$
Both outer groups vanish by parts a(iv) and a(v), so the <connecting homomorphism> is an isomorphism:
$$
\boxed{H^2(G,\mathbb Z)\cong H^1(G,\mathbb Q/\mathbb Z)}.
$$

Solution

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

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2021/iii/paper-151/4/b/ii/solution