On the outer automorphism groups of the absolute Galois groups of mixed-characteristic local fields (Q2164881)

Let \(p \in \mathbb{P}\) be a prime number, \(K/\mathbb{Q}_p\) a finite field extension and \(G_k\) the absolute Galois group of \(k\). Since the natural homomorphisms from \(\mathrm{Aut} (k)\), the automorphism group of \(k\), to \(\mathrm{Out} (G_k)\), the outer automorphism group of \(G_k\), is injective, one may regard \(\mathrm{Aut} (k)\) as a subgroup of \(\mathrm{Out} (G_k)\). The authors presuppose that \(p\) is odd and \(K/\mathbb{Q}_p\) is abelian with even degree. Using a result of Jannsen-Wingberg, see Theorem 7.5.14 in [\textit{J. Neukirch} et al., Cohomology of number fields. 2nd ed. Berlin: Springer. (2008; Zbl 1136.11001)], they prove that there exists some \(\alpha \in \mathrm{Out} (G_k)\), such that no power of \(\alpha\), when operating on the module \(k_+\), leaves the subset \(\mathbb{Q}_{p+}\) invariant. From this they deduce that \(\mathrm{Aut}(k)\) has infinitely many conjugates inside \(\mathrm{Out}(G_k)\), and therefore is not a normal subgroup.