# 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 ${\displaystyle H\subseteq S_n}$.