Suppose we have an abelian extension of Hopf algebras,
$$k \rightarrow k^G \rightarrow A \rightarrow kF \rightarrow k.$$
According to the general theory there is a left action of $F$ on $G$ and a $2$-cocycle $\sigma:F\times F \rightarrow k^G$ such that $A=k^G$ #${}_{\sigma} kF$ as algebras.
Is1)Is it true that $\sigma^n=\mathrm{id}$ for some $n\geq 1$? In other words does it followcan one choose $\sigma'$ in the same class with $\sigma$ such that if $\sigma_a(f,h)^n=\mathrm{id}$ if$\sigma'(f, h)=\sum_{a \in G}\sigma'_a(f,h)p_a$ then $\sigma(f, h)=\sum_{a \in G}\sigma_a(f,h)p_a$ where$$\sigma\;'_a(f,h)^n=\mathrm{id}$$ for all $a, f, h$. Here $p_a$ is the usual notation for dual basis of the group element basis .
- A little more general question (that implies the first question) is if the set of all $2$-cocycles of $F$ with values in $k^G$ finite? Equivalently the question is if $H^2(F, k^G)$ is finite with $F$ acting on $k^G$ via the action of $F$ on $G$.
A3)A related question, is the set of all for what groups $2$-cocycles of$X$ with an $F$ with values in-action the group $k^G$$H^2(F, X)$ is finite? For example $X=k^*$ gives the usual Schur multiplier.
