The Thompson–Wielandt theorem says that a point stabilizer in a finite primitive group is bounded, apart from one normal \(p\)-core, by a single subdegree. [@isaacs2008]
Definition
Core-free and maximal mean that the action is faithful and primitive respectively.
Core result
Analyze the intersection of \(H\) and \(H^g\) and its cores on both sides. When no nontrivial subgroup of the intersection is normal in a larger local group, iterated cores force one residual side to be a \(p\)-group; permutation embeddings yield the factorial bound.
Structural properties
- The prime \(p\) is not prescribed and emerges from the local intersection.
- The bound is coarse but depends only on subdegree \(m\), not on the full degree.
- A point stabilizer is therefore a bounded extension of a normal \(p\)-group.
- The theorem is an early local finiteness result for primitive groups.
- Core-free maximal-subgroup language is equivalent to faithful primitive action.
The result turns a local orbit size into a structural bound on a point stabilizer, a deep instance of permutation geometry controlling group order.
Example and boundary
Knowledge network
Primitivity gives the maximal stabilizer, suborbits explain \(m\), and the p-core is the large normal part in the conclusion.
Proof and sources
The precise theorem and the more general intersection-core version are Theorems 9.23-9.24. [@isaacs2008]
