= Bockstein on a cyclic Moore space
{c}
{title2=$\beta:H^i(M(\mathbb Z/n,i);\mathbb Z/n)\xrightarrow{\sim}H^{i+1}(M(\mathbb Z/n,i);\mathbb Z/n)$}
Attach an $(i+1)$-cell to $S^i$ by a degree-$n$ map, with $i\geq1$ and $n\geq2$. The integral cellular differential is multiplication by $n$, while its modulo-$n$ reduction is zero. Lifting the degree-$i$ generator modulo $n^2$ and applying the differential gives $n$, so the <Bockstein homomorphism> sends that generator to the degree-$(i+1)$ generator. This supplies a nonzero example in every positive degree.
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