A wreath product combines coordinate copies of one group with a second group permuting those coordinates. It is a universal container for extensions, a standard model for iterated permutation structure, and the natural obstruction in Yoshida's transfer theorem. [@isaacs2008]
Definition
When \(B\) acts regularly on itself, write \(A\wr B\) and call it the regular wreath product. The base \(A^X\) is normal and the top group \(B\) is a complement.
Core result
Choose a transversal for \(N\) in \(G\). For each \(g\in G\), record its kernel component at every quotient coset to form a function \(Q\to N\), and record the permutation induced by right multiplication. The multiplication law is exactly the wreath semidirect law, and the record determines \(g\).
Structural properties
- If \(A,B\) are finite and \(|X|=n\), then \(|A\wr_X B|=|A|^n|B|\).
- The imprimitive permutation action on blocks is the standard model for many imprimitive groups.
- Iterated wreath products describe automorphisms of rooted trees and recursive Sylow-subgroup structures.
- \(C_p\wr C_p\) has base \(C_p^p\), order \(p^{p+1}\), and nilpotency class \(p\).
- In Yoshida's theorem, \(C_p\wr C_p\) is the unique quotient obstruction to control of transfer by a Sylow normalizer.
A wreath product depends on a specified permutation action of the top group. Regular, product-action, and imprimitive versions should not be conflated.
Example and boundary
Knowledge network
Split extensions supply the semidirect formula, permutation groups explain coordinate blocks, and Yoshida's theorem displays the local obstruction represented by the regular \(p\)-wreath product.
Proof and sources
Construction and extension embedding are in Section 3C; the structure and transfer role of \(C_p\wr C_p\) are in Section 10A. [@isaacs2008]
