Let $p$ be a prime, what is the automorphism group of $G=\bigoplus_{i=-\infty}^{\infty}\mathbb{Z}_p$ (countable infinite direct sum of $\mathbb{Z}_p$)?
I know every permutation of the coordinates will be an automorphism, so $\mathrm{Sym}(\mathbb{Z})$ is a subgroup of ${\rm Aut}(G)$. I think there will be other automorphisms, because ${\rm Aut}\,\mathbb Z_p\simeq \mathbb Z_{p-1}$, so in each coordinate, we can map a generator to a different generator. Also, we can map a generator in one coordinate to a generator in another coordinate.
Any insight/reference would be really appreciated. Thanks.