# Endomorphisms of the p-adic group $(\mathbb Z_p,+)$

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 $$\mathbb Z_p$$?

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.