Let $G$ be a cyclic group of order $p$ and let $IG$ denote the augmentation ideal of the group ring $\mathbb{Z}G$. I need to find $H^1(G,IG)$.
Since
$$0 \rightarrow IG \rightarrow \mathbb{Z}G \rightarrow \mathbb{Z} \rightarrow 0$$
Then we get the long exact sequence
$$0 \rightarrow H^0(G,IG) \rightarrow H^0(G,\mathbb{Z}G) \rightarrow H^0(G,\mathbb{Z}) \rightarrow H^1(G,IG) \rightarrow 0$$
as $\mathbb{Z}G$ is projective.
I'm not sure how to get it using the above sequence. Is there an alternative way?
Thanks for help.