---
tag: W1X984PKJ9
kind: concept
name: Wreath product
msc: ["20E22"]
specializes: [0H3WZXCZNF]
sources:
  - url: https://en.wikipedia.org/wiki/Wreath_product
same_as:
  wikidata: Q1786640
  wikipedia: Wreath product
---

## Summary

The wreath product $A \wr_\Omega G$ builds a new group from a group $A$ and a group $G$ that acts on a set $\Omega$.
It is the [[semidirect product|0H3WZXCZNF]] of the group of all functions $\Omega \to A$ with $G$, where $G$ moves the functions by moving the points of $\Omega$.
Wreath products describe the symmetries of objects made of several copies of a smaller object.
Each [[group extension|MF4N3VGGFV]] $E$ of a group $B$ by a group $A$ embeds in the unrestricted regular wreath product $A \wr B$.

## Explanations

### L1 human

Think of $m$ identical boxes, each holding a copy of some object with symmetry group $A$.
A symmetry of the whole picture may turn each box by its own element of $A$, and then permute the boxes by an element of $G$.
The wreath product is the group of all such combined moves.
For example, take two coins on a table, each of which can be flipped, and allow the two coins to swap places.
Each coin gives a group of order $2$, the swap gives another group of order $2$, and the group of all moves is $C_2 \wr C_2$, of order $2^2 \cdot 2 = 8$.