Does there exist an endomorphism of $(\mathbb Z\_p,+)$ of finite order different of $x\mapsto\xi x$ where $\xi$ is a root of unity in $\mathbb Z\_p$?


Thanks in advance



Endomorphisms of the p-adic group $(\mathbb Z_p,+)$
I claim that any endomorphism of $\mathbb{Z}\_p$ is just multiplication by a constant.


Let $f$ be an endomorphism of $\mathbb{Z}\_p$. First observe that $p^n f(x) = f(p^n x)$, so $f$ takes $p^n \mathbb{Z}\_p$ to $p^n \mathbb{Z}\_p$. This shows that $f$ is continuous. For any $x \in \mathbb{Z}\_p$, we can find a sequence of integers $x\_1, x\_2, \ldots$ such that $x\_n \to x$. By continuity, we have $x\_n f(1) = f(x\_n) \to f(x)$ and therefore $f(x) = x f(1)$.


If $f$ moreover has finite order, then $f(1)$ has to be a root of unity. 


