The automorphism group of Prüfer group

Question

Let $G = C_{p^\infty}$ be the Prüfer group, where $p$ is a prime. Is clearly that $G$ is a torsion group.

In general, if $G$ is a torsion group, it is not true that $Aut(G)$ is a torsion group.

But in the specific case where $G = C_{p^\infty}$, is true that $Aut(G)$ is a torsion group?

Remark: $Aut(G)$ is the group of automorphisms of $G$.

Answer 1

First note that the endomorphism ring of $G$ is the ring $A$ of $p$-adic integers.

Its groups of units (which is of course isomorphic to the authomorphism group you are interested in) has a subgroup of index $p-1$ which is isomorphic to the additive group of $A$, so definitely not torsion.