After fixing one point, the orbits of its stabilizer are the suborbits. They record the local geometry that remains after one level of stabilization. [@isaacs2008]
Definition
The trivial suborbit \(\{\alpha\}\) has subdegree \(1\). The multiset of subdegrees is independent of the chosen point.
Core result
If \((\alpha,\beta)\) and \((\alpha,\gamma)\) lie in one orbital, the transporting element fixes \(\alpha\), so \(\beta,\gamma\) are in one \(G_\alpha\)-orbit. The converse is immediate.
Structural properties
- Rank \(2\) is equivalent to double transitivity.
- Paired orbitals give reverse suborbits and have equal subdegree.
- A subdegree is the out-valency of its orbital digraph.
- Nontrivial subdegrees of primitive groups satisfy strong divisibility and growth restrictions.
- The largest subdegree is a local index in the point stabilizer.
Suborbits compress global ordered-pair orbits into a point-stabilizer problem and form the interface between permutation groups and graph theory.
Example and boundary
Knowledge network
Orbital graphs turn suborbits into edges, primitivity gives connectivity, and the permutation core article develops orbitals.
Proof and sources
Orbital-suborbit correspondence and subdegree restrictions are in Section 8D. [@isaacs2008]
