Finite-Orbit Actions and Exact Reconstruction

Abstract

We associate a profinite group to every group \(G\) acting on a set \(\Omega\) with finite orbits. For each finite \(G\)-stable subset \(A\subseteq\Omega\), 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 $\Gamma_\Omega:=\varprojlim_A G_A$. We show that \(\Gamma_\Omega\) acts naturally on \(\Omega\) and is canonically topologically isomorphic to the closure of the image of \(G\) in \(\operatorname{Sym}(\Omega)\), endowed with the topology of pointwise convergence. We introduce the finite-level exactness property \(\textup{FLEP}\), under which subgroups of \(\Gamma_\Omega\) 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 \(\Gamma_\Omega\) and the fixed subsets of \(\Omega\) arising from closed subgroups. We apply the theory in two directions. First, every profinite group \(\Gamma\) is recovered from its normal finite-quotient action on $\coprod_{N}\Gamma/N$, where \(N\) ranges over the open normal subgroups of \(\Gamma\). For this action, \(\textup{FLEP}\) holds precisely when every finite quotient \(\Gamma/N\), 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 $\Gamma_E \cong_{\mathrm{top}} \operatorname{Gal}(E/F)$, where \(\operatorname{Gal}(E/F)\) has the Krull topology. Thus the Krull Galois group is recovered from finite-orbit data.