Overview
This work establishes a construction linking a group $G$ acting on a set $\Omega$ with finite orbits to a specific profinite group, denoted $\Gamma_\Omega$. The construction involves forming an inverse system of induced finite permutation groups and defining $\Gamma_\Omega$ as its inverse limit. The action of $\Gamma_\Omega$ on $\Omega$ is shown to be natural, and $\Gamma_\Omega$ is canonically topologically isomorphic to the closure of the image of $G$ in $\operatorname{Sym}(\Omega)$, considering the topology of pointwise convergence.
Approach
The core of the approach involves defining $\Gamma_\Omega$ through an inverse limit. For a group $G$ acting on $\Omega$ with finite orbits, consider every finite $G$-stable subset $A\subseteq\Omega$. For each such $A$, $G_A\leq\operatorname{Sym}(A)$ represents the induced finite permutation group. These groups $G_A$, along with their natural restriction maps, constitute an inverse system. The profinite group $\Gamma_\Omega$ is then defined as the inverse limit of this system, i.e., $\Gamma_\Omega:=\varprojlim_A G_A$.
The study introduces the finite-level exactness property (FLEP). Under FLEP, subgroups of $\Gamma_\Omega$ can be recovered up to closure from their fixed-point sets, and closed subgroups are exactly recovered. The research provides several equivalent formulations for FLEP.
Findings
- The constructed group $\Gamma_\Omega$ acts naturally on $\Omega$.
- $\Gamma_\Omega$ is canonically topologically isomorphic to the closure of the image of $G$ in $\operatorname{Sym}(\Omega)$, under the topology of pointwise convergence.
- Under FLEP, the fixed-point set construction establishes an inclusion-reversing bijection between closed subgroups of $\Gamma_\Omega$ and the fixed subsets of $\Omega$ that originate from closed subgroups.
- When FLEP holds, subgroups of $\Gamma_\Omega$ are recovered up to closure from their fixed-point sets, and closed subgroups are recovered precisely.
Applications
The developed theory is applied in two distinct directions:
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Profinite Group Reconstruction: Every profinite group $\Gamma$ can be reconstructed from its normal finite-quotient action on $\coprod_{N}\Gamma/N$. Here, $N$ ranges over the open normal subgroups of $\Gamma$. For this specific action, FLEP holds if and only if every finite quotient $\Gamma/N$ (where $N$ is open and normal) is a Dedekind group.
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Galois Group Reconstruction: Consider a scenario where $G\leq\Aut(E)$ acts on a field $E$ with finite orbits, and $F=E^G$ is the fixed field. Under these conditions, the field extension $E/F$ is Galois. The construction yields a canonical topological isomorphism, $\Gamma_E \cong_{\mathrm{top}} \operatorname{Gal}(E/F)$, where $\operatorname{Gal}(E/F)$ is endowed with the Krull topology. This demonstrates that the Krull Galois group can be recovered from finite-orbit data.