Remark

Let $p > 2$ be a prime number and let $x$ be an integer congruent to 1 mod $p$ and not congruent to 1 mod $p^2$; there is an exact sequence

of groups, where $v$ is the natural projection and where $u$ is induced on the quotients by the map $r \mapsto x^r$. Let $\mathbf{Z}_p$ be the ring of $p$-adic integers ($V$, p. 96), and let $x$ be an element of $\mathbf{Z}_p$ such that $x - 1 \in p\mathbf{Z}_p$ and $x - 1 \notin p^2\mathbf{Z}_p$; on passing to inverse limits, the exact sequences (5) induce an exact sequence

where $v$ is the natural map, and the continuous map $u$ extends the map $n \mapsto x^n$ ($n \in \mathbf{Z}$). We often put $x^n = u(x)$ for $n \in \mathbf{Z}_p$.

Similarly, if $x \in \mathbf{Z}_2$ with $x - 1 \in 4\mathbf{Z}$, and $x - 1 \notin 8\mathbf{Z}_2$, then there is a split exact sequence

where $u$ is a continuous extension of the map $n \mapsto x^n$