n-singular chains are result of a group action?

Question

I´m beginning in the study of Singular Homology using Kosniowski's text.

Since it seems that the n-chains group $S_n(X)$ inherites the group structure of $\mathbb{Z}$, we can consider that, in certain sense, exist a group action $$\cdot : \mathbb{Z} \times S_n(X) \to S_n(X)$$ such that $$(n_j,\sum_j\varphi_j) := \sum_j n_j \varphi_j$$ for a finite number of non-zero $n_j \in \mathbb{Z}$.

Is it right this "idea"?

Thanks!