Finite-Orbit Actions and Exact Reconstruction

Abstract: We associate a profinite group to every group (G) acting on a set (Ω) with finite orbits. For each finite (G)-stable subset (A\subseteqΩ), let (G_A\leq\operatorname{Sym}(A)) be the induced finite permutation group. The groups (G_A), with the natural restriction maps, form an inverse system, and we define $ΓΩ:=\varprojlim_A G_A$. We show that (ΓΩ) acts naturally on (Ω) and is canonically topologically isomorphic to the closure of the image of (G) in (\operatorname{Sym}(Ω)), endowed with the topology of pointwise convergence. We introduce the finite-level exactness property (\textup{FLEP}), under which subgroups of (ΓΩ) are recovered up to closure from their fixed-point sets, and closed subgroups are recovered exactly. We prove several equivalent formulations of (\textup{FLEP}). Under this condition, the fixed-point set construction gives an inclusion-reversing bijection between closed subgroups of (ΓΩ) and the fixed subsets of (Ω) arising from closed subgroups. We apply the theory in two directions. First, every profinite group (Γ) is recovered from its normal finite-quotient action on $\coprod_{N}Γ/N$, where (N) ranges over the open normal subgroups of (Γ). For this action, (\textup{FLEP}) holds precisely when every finite quotient (Γ/N), with (N) open and normal, is a Dedekind group. Second, if (G\leq\Aut(E)) acts on a field (E) with finite orbits and (F=E^G), then (E/F) is Galois and the construction yields a canonical topological isomorphism $ΓE \cong{\mathrm{top}} \operatorname{Gal}(E/F)$, where (\operatorname{Gal}(E/F)) has the Krull topology. Thus the Krull Galois group is recovered from finite-orbit data.