In Example 1B.7 of Hatcher's 'Algebraic Topology', he attempts to construct a $K(G,1)$ space for any group G.
His construction goes as follows: Consider the $\Delta$-complex (denote it as $EG$) formed by n-simplices $[g_0,...,g_n]$ where $g_0,...,g_n$ are elements of $G$, with the n-simplices attaching to (n-1)-simplices in the natural way. Then he defines an action of $G$ on $EG$ by: $g \in G$ takes $[g_0,...,g_n]$ linearly onto $[gg_0,...,gg_n]$. Then he claims: this action is a covering space action (that is, for each $x \in EX$, there exists a neighborhood $U$ such that the images $g(U)$, $g\in G$ are disjoint) (See also: How to build a $K(G,1)$ space for every group $G$?, this post may describe the setting of the construction better)
I cannot see how this claim can hold; consider the case when $G=\mathbb{Z}_4$ (written additively), then consider the midpoint of the 1-simplex $[0,2]$, it will be mapped to itself under the action of $2\in G$, so certainly the condition for covering space action is not met?
Can anyone please point out to me what I missed here? Any help is appreciated.