= Abelianization over a Zp-extension
{title2=$\mathcal G^{\mathrm{ab}}:\quad X_\Gamma=X/(\gamma-1)X$}
Suppose $1\to X\to\mathcal G\to\Gamma\to1$ is an exact sequence of <pro-p groups>, with $X$ abelian and $\Gamma\cong\mathbb Z_p$. Conjugation gives $X$ a <compact Galois module> structure. The closed <commutator subgroup> is $(\gamma-1)X$: its image is closed by compactness, and after quotienting by it, a lift of $\gamma$ centralizes $X$ and topologically generates the remaining quotient. Hence $0\to X_\Gamma\to\mathcal G^{\mathrm{ab}}\to\Gamma\to0$ is exact.
Back to article page
