= Periodic group cohomology from a free sphere action
{title2=$\Delta\in H^n(BG;\mathbb F)$}
Suppose a finite group acts freely on $S^{n-1}$ and trivially on its integral <cohomology>. The <Borel construction> gives a sphere fibration over $BG$ whose total space is homotopy equivalent to the $(n-1)$-dimensional quotient. Its <Serre spectral sequence> has just two rows with trivial local coefficients. Transgressing the fibre generator gives $\Delta$; the sole differential is multiplication by it up to sign. For $i>0$, both the source and target terms have total degrees above $n-1$, so their surviving kernel and cokernel must vanish. Hence multiplication by $\Delta$ gives period $n$ in positive-degree cohomology over any field.
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