Let $k$ be a $p$-adic local field with absolute Galois group $G_k$. In Cohomology of Number Fields, the authors define the $n$-th Tate twist of a finite $G_k$-module $A$ as the $G_k$-module $A(n)$ that is equal to $A$ as an abelian group and endowed with the twisted action $\sigma(a) = \chi(\sigma)^n \cdot \sigma a$, where $\chi$ denotes the cyclotomic character and the action on the right-hand side is the original action of $G_k$ on $A$.
I do not quite understand how the multiplication of $\chi(\sigma)^n$ with $\sigma a$ works. $\chi$ takes values in some product $\prod \mathbb{Z}_l^\times$, so how can this be multiplied with the elements of an arbitrary finite $G_K$-module $A$? More concretely, I would like to understand the modules $\mathbb{Z}/n\mathbb{Z}(i)$. It is claimed that $\mathbb{Z}/n\mathbb{Z}(i) \cong \mu_n^{\otimes i}$, but I do not quite understand the cyclotomic action in this case.