I am learning class field theory from the famous ANT book by Neukirch, where I stuck at the middle of a long proof, whose goal is to prove multiplicativity of reciprocity map.
Defining various notations here costs tremendous space, so non-readers of Neukirch may have trouble understanding this. The precise location is at Chapter 4, section 5, bottom of page 292 and top of 293:
It is evident that $\tilde{u}^{\varphi-1} \in U_{\Sigma_n}$ because $\sigma^n$ and $\varphi$ commutes, so $(\tilde{u}^{\varphi-1})^{\sigma^n} = \tilde{u}^{\sigma^n (\varphi-1)} = \tilde{u}^{\varphi-1}$.
On the other hand, it is not clear why $\tilde{u_i}^{\tau_i-1} \in U_{\Sigma_n}$. One might attempt to show it is fixed by $\sigma^n$: but the problem comes as $\tau_i$ and $\sigma^n$ need not commute. From another direction, I also try to show some extension of form $\Sigma_n / E$, where $E\subset \tilde{K}$ is Galois, this would also prove $\tilde{u_i}^{\tau_i-1} \in U_{\Sigma_n}$. But so far, neither approach have been successful.
I have read the rest of the proof but they make no indication why this is true. So my question is:
Why is $\tilde{u_i}^{\tau_i} \in U_{\Sigma_n}$ ?
Neukirch seems to manipulate these things as if $G$ were abelian. Perhaps I am missing something very obvious, this question has been baffling me for a day. Thank you very much.