The orbit-stabilizer theorem links geometric motion in a group action exactly to subgroup index. It is the common counting mechanism behind conjugacy-class sizes, Sylow arguments, Burnside orbit counting, and permutation representations. [@isaacs2008]
Definition
This article uses right-action notation \(\alpha^g\). Under a left action, right cosets are replaced by left cosets; the index formula is unchanged.
Core result
The equality \(\alpha^g=\alpha^h\) holds exactly when \(gh^{-1}\in G_\alpha\), proving both well-definedness and injectivity; the definition of orbit gives surjectivity. Isaacs, Theorem 1.4 strengthens the bijection to an isomorphism of permutation actions.
Structural properties
- Orbits partition \(\Omega\), so the size of a finite \(G\)-set is the sum of its orbit sizes.
- The action is transitive exactly when there is one orbit, in which case \(|\Omega|=|G:G_\alpha|\).
- It is regular exactly when it is transitive and \(G_\alpha=1\), giving \(|G|=|\Omega|\) in the finite case.
- For conjugation on elements, \(G_x=C_G(x)\), so the class of \(x\) has size \(|G:C_G(x)|\).
- For conjugation on subgroups, the stabilizer of \(H\) is \(N_G(H)\), so the number of conjugates is \(|G:N_G(H)|\).
The formula converts stabilizer size into orbit size and conversely turns a small orbit into a large stabilizer. Finite-group proofs repeatedly move between these two viewpoints.
Example and boundary
Knowledge network
The core group-action article supplies kernels, fixed points, and examples. The class-equation and permutation-group articles develop the conjugation and transitive-action applications.
Proof and sources
The coset correspondence and conjugation consequences are Isaacs, Theorems 1.4-1.6; the short proof is retained here and later uses are developed in the linked articles. [@isaacs2008]
