In a paper by Farkas, I was doing this lemma, where I had this doubt (red underlined) in the proof of the lemma.
Can anybody explain me how does it follow $\alpha$ is centralized by $H$. It should mean that $h^{-1}\alpha h=\alpha\ \forall\ h \in H$, but I don't see it.
$U_1(ZG)$ here means units of $ZG$ with augmentation $1$, i.e. $u=\sum_{g\in G} a(g)g \in U(ZG)$ such that $\sum a(g)_{g \in G}=1$
Here the argument isn't that $\alpha$ is centralized by $H$, but rather that $\alpha$ is centralized by a subgroup of finite index in $H$, which in turn must be of finite index in $G$. This follows because the index of the centralizer in $H$ of $\alpha$ is the same as the order of $\alpha$'s orbit under the conjugation action of $H$, by the orbit-stabilizer theorem.