I got stuck in J.S.Milne's Class Field Theory Chapter 2, Page 79. It is written in the remark that for a cyclic group $G$ with generator $\sigma$.
Let $\gamma$ be the element of $H^2(G,\mathbb{Z})$ corresponding under the isomorphism
$H^2(G,\mathbb{Z})\simeq Hom(G,\mathbb{Q/Z})$
to the map sending the chosen generator $\sigma$ of $G$ to $1/m$. Then the map
$H^r_T(G,M)\to H^{r+2}_T(G,M)$ is just $x \mapsto x\cup\gamma$.
I tried to show this by definition but failed. I have to go back to the exact sequence
$0\to M\to \mathbb{Z}[G]\otimes_\mathbb{Z}M\to \mathbb{Z}[G]\otimes_\mathbb{Z}M\to M\to 0$
in the previous page but found it hard to compute with the cohomology $\delta$ functor. By the way, I also have no idea about how to show a homomorphism is just the cup product with this given $\gamma$. So does anyone knows how to deal with this problem?