Let $G$ be a finite group. Define $d_1:Z[G]\to Z[G]$ morphism by the following.
Take basis element $\sigma\in G$. Define $d_1(\sigma)=1-\sigma$. Then $d_1$ is a $G-$module morphism.
$\textbf{Q:}$ I tried to verify $g\in G$ $d_1(g\sigma)=gd_1(\sigma)$. However, I got $d_1(g\sigma)=1-g\sigma$ whereas $gd_1(\sigma)=g(1-\sigma)$. I cannot even match the basis in the image. Have I done something wrong here? I would expect this definition should give me $G-$module morphism as I was trying to check standard free resolution of $Z$ is a complex. And I am stucked at the first step.
Ref: Neukirch Bonn Lectures Chpt 1. Sec 2.