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A group coboundary is a cochain in the image of the preceding coboundary map. Quotienting group cocycles by group coboundaries gives group cohomology.

A group cocycle is an element of the kernel of the coboundary map in the standard cochain complex computing group cohomology.

The natural maphas finite image by hypothesis. It remains to bound its kernel. If $P\in E(K)$ becomes $nQ$ for $Q\in E(L)$, thenis a one-cocycle for $G=\mathrm{Gal}(L/K)$. Changing $Q$ by an $n$-torsion point changes this cocycle by a coboundary, producing a well-defined map from the kernel toIf its cohomology class is zero, subtracting the corresponding torsion point from $Q$ makes $Q$ Galois fixed, so $P\in nE(K)$. The map is therefore injective. Both $G$ and $E(L)[n]$ are finite, so this group cohomology set is finite. A finite kernel and finite image give

Choose a projective resolution $P_{\bullet }\to \mathbb{Z}$ of the trivial $\mathbb{Z}G$-module. For a $\mathbb{Z}G$-module $M$, group cohomology isDifferent projective resolutions give naturally isomorphic groups.