Let $G := \underbrace{\mathbb{Z}/p^{e}\mathbb{Z} \times \ldots \times \mathbb{Z}/p^{e}\mathbb{Z}}_{n} = \{(a_{1},\ldots,a_{n}) : a_{i} \in \mathbb{Z}/p^{e}\mathbb{Z} \}$, and consider the action by $\text{Sym}(n)$ that permutes the factros. That is, for some $\sigma \in \text{Sym}(n)$,
$(a_{1},\ldots,a_{n})\sigma = (a_{\sigma(1)},\ldots,a_{\sigma(n)}).$
So $\sigma$ defines a map between $G$ and itself. I have two questions about this.
Is this action bijective? I believe it is but my calculations are getting me nowhere
Is this a homomorphism? I also have a feeling that it is. If this is true, does this mean that $\text{Sym}(n) \leq \text{Aut}(G)$?