$H^1(G,M)$ be group cohomology. This group depends on how $G$ acts on $M$. But I'm interested in the maximum valu of this group.
What is the maximum value of $\sharp H^1(\Bbb{Z}/2\Bbb{Z},\Bbb{Z}×\Bbb{Z}/2\Bbb{Z})$?
When the action is trivial,$\sharp H^1(\Bbb{Z}/2\Bbb{Z},\Bbb{Z}×\Bbb{Z}/2\Bbb{Z})=0$. If $\Bbb{Z}/2\Bbb{Z}$ acts differently, the value may differ, but what is the maximum value of them ?