Let $A$ be a finite $\mathbb{Z}$-module (i.e., a finite abelian group). My question is: for what $n\in \mathbb{Z}^{n\geq 2}$ the map \begin{align} \alpha_{n}:\bigwedge^nA&\to A^{\otimes n}\\ a_1\wedge \cdots \wedge a_n&\mapsto \sum_{\pi\in \mathbb{S}_n}(sig(\pi))a_{\pi(1)}\otimes \cdots\otimes a_{\pi(n)} \end{align} is injective? For $n=2$, $\alpha_2$ is injective. It follows, for example, just by induction on $r$, if $A=\mathbb{Z}/\mathbb{Z}m_1\oplus \cdots \oplus\mathbb{Z}/\mathbb{Z}_{m_r}$.
Thanks!