I'm having a little perplexity about the followoing fact explained in this paper (Example 1.1):
Consider the group $G :=\mathbb{Z}_6$ and define its action on $\textbf{x} \in \mathbb{R}^3$ by defining it on the index set $N=\{1,2,3\}$ as $g \cdot n := g+n \, \, \pmod{3}\, \, \, \, \forall g \in \mathbb{Z}_6$. Is this a well-defined action on $\textbf{x} = (x_1,x_2,x_3)$?
Let's take $g=1$, and let it act on $N$. Then we would have
$$\textbf{x}=(x_{g\cdot1}, x_{g \cdot 2}, x_{g \cdot 3}) = (x_2, x_0,x_1)$$
What should $x_0$ correspond to here?
Is this just a notational issue or is it a-priori wrong to consider $\pmod{3}$ addition on $\mathbb{Z}_6$?
It seems like you know what they intended to say, ie. the group acts by permuting the coordinates. Strictly speaking, their notation is wrong (or at the very least pretty confusing).
Quick fix: set $N = \{0, 1, 2\}$, and define the action in the same way, where we view $x \in \mathbb{R}^3$ as $x = (x_0, x_1, x_2)$. This will result in the 'same' action without the notational errors you noticed.