I am working through some exercises in Neukirch's Algebraic number theory and i need some hints for exercise 1 pg 274, it goes as follows:
Let $G$ be a profinite group, show that we can extend the power map $G \times \mathbb{Z} \rightarrow G$, $(\theta,a) \mapsto \theta^{a}$, to a continuous map $G \times \hat{\mathbb{Z}} \rightarrow G$, $(\theta,a) \mapsto \theta^{a}$, and that $\theta^{ab} = (\theta^{a})^{b}$, $\theta^{a+b}=\theta^{a}\theta^{b}$ if $G$ is abelian.
Now my idea to extend the map was the following, since $G$ is profinite we have $G \cong \varprojlim G/N_{i}$ where the limit runs over the $open$ normal subgroups $N_{i}$ of $G$ , and we know $\hat{\mathbb{Z}} \cong \varprojlim \frac{\mathbb{Z}}{n\mathbb{Z}}$, so let $\theta \in G$, and define $\theta_{i} = \theta \mod N_{i}$, and for $a \in \hat{\mathbb{Z}}$ do a similar thing, then I defined $\theta^{a} = (\theta_{i}^{a_{i}})_i$, now im not sure if this is correct or not, and even if it is right i am not quite sure how to prove its continuous, this is where I need some hints.
Thank you
Correction: Actually, thinking about it a but more I think my way of extending the map might not work, beacuse i dont know that every profinite has enough normal subgroups in order to make the definition work, maybe I shoud define $\theta^{a} = \prod \theta^{a_{i}} $, but Im not sure how to prove its continuous.