Calculations of some groups of Hopf algebra extensions (Q1357801)

It is known that for a matched pair \((G,F)\) of two subgroups \(F\) and \(G\) of a group \(L\), \(\text{Opext}(kF,k^G)\) denotes the set of the equivalence classes of the extensions of \(kF\) by \(k^G\), which forms an abelian group. Let \(C_n\) denote the cyclic group of order \(n\) and \(S_n\) the symmetric group of degree \(n\). The purpose of this paper is to calculate the Opext groups for three matched pairs with some restrictions on the ground field \(k\). It is shown that (i) for the matched pair \((C_n\times C_n,C_2)\), \(\text{Opext}(kC_2,k^{C_n\times C_n})\cong\mu_n\) where \(\mu_n\) is the group of all \(n\)th roots of 1 contained in \(k\); (ii) for the matched pair \((C_n\times C_n,C_n)\), \(\text{Opext}(kC_n,k^{C_n\times C_n})\cong\begin{cases}\mu_n\times\mu_n, &\text{if }n\) is odd