Thompson replacement is a method rather than one isolated formula. It moves a maximum-order abelian subgroup into a position with a larger normalizer or better centralization while preserving its order. [@isaacs2008]
Definition
\(\mathcal A(P)\) is the set of maximum-order abelian subgroups of \(P\), and the replacement must remain in that set.
Core result
A commutator pairing embeds \(A/(A\cap B)\) in a group controlled by \(B/C_B(A\cap B)\), giving \(|A^*|\geq|A|\). Maximality forces equality, and a strictly increasing finite invariant makes the process terminate.
Structural properties
- Replacement preserves maximum order but does not assert that \(A^*\) is conjugate to \(A\).
- The class-two condition makes the commutator pairing bilinear and protects abelianness.
- It is used to place \(J(P)\) under a local control group's normalizer.
- Typical monotone quantities are intersection size, normalizer order, or fixed-point dimension.
- Precise hypotheses and Thompson-subgroup conventions differ across sources.
When maximum abelian subgroups are nonunique, replacement strengthens existence to existence in a strategically useful position.
Example and boundary
Knowledge network
The Thompson subgroup collects all maximum abelian subgroups, coprime action supplies external control, and the normal-complement theorem is the main output.
Proof and sources
Replacement arguments drive Sections 7A-7C; exact forms should be cited to the theorem and lemmas used there. [@isaacs2008]
