# Wreath product

Wreath product
Symbol${\displaystyle \wr }$[1]
Size formula${\displaystyle |G\,{\wr }_{n}H|=|G|^{n}|H|}$[2]
Algebraic properties
Algebraic structureUnital magma
AssociativeNo
CommutativeNo
IdentityS0

The wreath product is an product that operates on groups.

## Definition

Given a group ${\displaystyle (G,\times )}$ and a permutation group ${\displaystyle (H,*)}$ such that ${\displaystyle (H,*)\subseteq S_{n}}$, then ${\displaystyle G\wr H}$ is a group ${\displaystyle (K,\cdot )}$ where:

${\displaystyle K=\left\{(g,h)\mid g\in G^{n},h\in H\right\}}$

and:

${\displaystyle ((g_{1},\dots ,g_{n}),h)\cdot ((g_{1}',\dots ,g_{n}'),h')=((g_{h'(1)}\times g_{1}',\dots ,g_{h'(n)}\times g_{n}'),h*h')}$

Put simply the operation permutes the tuple g  in the first argument by the permutation h'  in the second argument, and then combines the two tuples pairwise with the usual operations.[1]

This definition can be generalized to apply to arbitrary groups, by observing that due to Cayley's theorem every group is isomorphic to a permutation group.

## Properties

• For groups G  and ${\displaystyle H\subseteq S_{n}}$: ${\displaystyle |G\wr H|=|G|^{n}|H|}$.
• If ${\displaystyle G\subseteq S_{m}}$ and ${\displaystyle H\subseteq S_{n}}$ then ${\displaystyle G\wr H}$ is isomorphic to a subgroup of Smn .[1]

## References

1. McMahan, Peter (2003). Wreath-Product Polytopes (PDF) (Thesis). Reed College.
2. Here n  is determined by ${\displaystyle H\subseteq S_{n}}$.