``Hasse principle'' for \(PSL_2(\mathbb{Z})\) and \(PSL_2(\mathbb{F}_p)\) (Q1281523)

Let \(g\) be a group which acts on another group \(G\), let \(h\leq g\). Then there is a restriction map \(r_h: H^1(g,G)\to H^1(h,G)\) of 1-cohomology sets. Let now \({\mathcal H}\) be a family of subgroups \(h\leq g\). Put \(\text{ Ш}_{\mathcal H}(g,G)= \bigcap_h \ker(r_h)\), \(h\in{\mathcal H}\). The author calls this a ``Tate-Shafarevich set''. In the paper he studies the case \(g=G\) (where \(G\) acts by inner automorphisms on itself), \({\mathcal H}\) the family of all cyclic subgroups of \(G\), and he writes \(\text{ Ш}(G)\) instead of \(\text{ Ш}_{\mathcal H} (G,G)\). He proves \(\text{ Ш}(G)=1\) for the groups in the title.