# Conjugation action of $Gal(\bar{s}/s)$ on the tame ramification group

There is a statement in SGA 7-1 Exposé 1 (P. Deligne, Résumé des premiers exposés de A. Grothendieck, pdf of SGA7-1), (0.3.1):

$$S$$ is a Henselian trait (i.e. the spectrum of a henselian discrete valuation ring), $$\eta$$, $$s$$, $$V$$, $$\bar{V}$$, $$\bar{\eta}$$, $$\bar{s}$$ as usual, and $$1\to I\to Gal(\bar{\eta}/\eta) \to Gal(\bar{s}/s) \to 1$$. $$p$$ is a characteristic of the residue field of $$s$$, $$P$$ be the pro-$$p$$ part of $$I$$, then the tame ramification group $$I/P$$ is canonically isomorphic to $$\hat{\mathbb{Z}}(1)(k(\bar{s}))$$.

Then the action of $$Gal(\bar{s}/s)$$ on $$I/P$$ can be identified with the natural action of $$Gal(\bar{s}/s)$$ on $$\hat{\mathbb{Z}}(1)(k(\bar{s}))$$.

It refers to Serre's Local Fields for proofs, but I cannot come up with a proof of this statement out of the this reference. Can somebody elaborate on this, or refer to a relevant section of the book?