Thompson's theorem compresses the many local normalizer tests in Frobenius's criterion to two canonical objects of a Sylow subgroup: its center and its Thompson subgroup. [@isaacs2008]
Definition
Several Thompson-subgroup conventions occur in the literature; this theorem uses the maximum-order abelian version.
Core result
In a minimal counterexample, replacement moves a local obstruction into either a central case or a maximum-abelian case. These lie in \(C_G(Z(P))\) and \(N_G(J(P))\) respectively, and the two hypotheses eliminate both. See the core article and Theorem 7.1.
Structural properties
- This form is an odd-prime theorem; additional local obstructions occur at \(p=2\).
- A stronger version tests normalizers of every nontrivial characteristic subgroup of \(P\).
- Characteristicity of \(J(P)\) makes its normalizer canonical.
- The theorem is used to prove nilpotence of an arbitrary Frobenius kernel.
- Glauberman ZJ theory later uses \(Z(J(P))\) under stability hypotheses.
The center and maximum abelian subgroups represent two extreme local structures in a \(p\)-group, and replacement forces every obstruction into one of them.
Example and boundary
Knowledge network
The Thompson-subgroup core article explains replacement, Frobenius's criterion supplies the global background, and Frobenius groups give the principal application.
Proof and sources
The characteristic-subgroup version is Theorem 6.23; the refined center-and-J(P) statement and proof are Theorem 7.1. [@isaacs2008]
