Compact groups having almost discrete orbit hypergroups (Q1364383)

Let \(G\) be a compact group of automorphisms acting continuously on a compact group \(H\), let \(H^G\) be the space of all \(G\)-orbits in \(H\) with natural hypergroup structure, let \(\mathbb{Z}(p)\) be a cyclic group and \(\mathbb{Z}_p\) the group of \(p\)-adic integers (\(p\) a prime). The author proves: (1) Let \(H\) be a compact group such that the orbit hypergroup \(H^G\) is almost discrete for some compact subgroup \(G\) of \(\Aut(H)\). If \(H\) contains an open solvable subgroup \(L\), then \(H\) is isomorphic either to \(\mathbb{Z}^n_p\) or to \(\mathbb{Z}(p)^N\) for some prime \(p\in N\). (2) Let \(H=\mathbb{Z}^n_p\), then \(H^G\) is almost discrete if and only if \(G\) is essentially an open subgroup of \(\text{CL}_n(\mathbb{Z}_p)\). -- Some interesting examples are also considered.