wreath product

Let $A$ and $B$ be groups, and let $B$ act on the set $\mathrm{\Gamma }$. Define the action of $B$ on the direct product $A^{\mathrm{\Gamma }}$ by

$$bf(\gamma ):=f(b^{-1}\gamma ),$$

for any $f\in A^{\mathrm{\Gamma }}$ and $\gamma \in \mathrm{\Gamma }$. The wreath product of $A$ and $B$ according to the action of $B$ on $\mathrm{\Gamma }$, denoted $A{\wr }_{\mathrm{\Gamma }}B$, is the semidirect product of groups $A^{\mathrm{\Gamma }}⋊B$.

Let us pause to unwind this definition. The elements of $A{\wr }_{\mathrm{\Gamma }}B$ are ordered pairs $(f,b)$, where $f\in A^{\mathrm{\Gamma }}$ and $b\in B$. The group operation is given by

$$(f,b)(f^{\prime },b^{\prime })=(fb{f}^{\prime },b{b}^{\prime }).$$

Note that by definition of the action of $B$ on $A^{\mathrm{\Gamma }}$,

$$(fb{f}^{\prime })(\gamma )=f(\gamma )f^{\prime }(b^{-1}\gamma ).$$

The action of $B$ on $\mathrm{\Gamma }$ in the semidirect product permutes the elements of a tuple $f\in A^{\mathrm{\Gamma }}$, and the group operation defined on $A^{\mathrm{\Gamma }}$ gives pointwise multiplication. To be explicit, suppose $\mathrm{\Gamma }$ is an $n$-tuple, and let $(f,b),(f^{\prime },b^{\prime })\in A{\wr }_{\mathrm{\Gamma }}B$. Let $b_i$ denote $b^{-1}(i)$. Then

$(f,b)(f^{\prime },b^{\prime })$ $=$ $((f(1),f(2),\mathrm{\dots },f(n)),b)((f^{\prime }(1),f^{\prime }(2),\mathrm{\dots },f^{\prime }(n)),b^{\prime })$

$=$ $((f(1),f(2),\mathrm{\dots },f(n))(f^{\prime }(b_1),f^{\prime }(b_2),\mathrm{\dots },f^{\prime }(b_n)),b{b}^{\prime })\text{(*)}$

$=$ $((f(1)f^{\prime }(b_1),f(2)f^{\prime }(b_2),\mathrm{\dots },f(n)f^{\prime }(b_n)),b{b}^{\prime }).$

Notice the permutation of the indices in (*).

A bit amount of thought to understand this slightly messy notation will be illuminating, and might also shed some light on the choice of terminology.

Title	wreath product
Canonical name	WreathProduct
Date of creation	2014-12-30 11:23:20
Last modified on	2014-12-30 11:23:20
Owner	mps (409)
Last modified by	juanman (12619)
Numerical id	21
Author	mps (12619)
Entry type	Definition
Classification	msc 20E22
Defines	wreath product