Interpretation of wreath products in general and on symmetric groups

Question

I've been trying to study how wreath products work. In several textbooks, K wr H is defined as the semidirect product of H acting on the set of all functions from X to K. Now, in my understanding, the semidirect product is just the direct product with a conjugation-like definition of multiplication. I cannot seem to connect the two concepts. Moreover, I have been told that a practical explanation of wreath products would be as such: Consider 3 pairs of shoes, with one pair on each row of the shoe rack. Taking the wreath product would be permuting the positions of the 3 shoes and permuting each pair.

Any help would be much appreciated. Thank you.

Answer 1

First if $X$ is a non-empty set and $K$ is a group, then $\mathcal{F}(X,K)$ (the set of functions from $X$ to $K$) is clearly a group for the following law : $$f*g:x\mapsto f(x)*_Kg(x)\text{.} $$

If $H$ is a group naturally acting on $X$, then $H$ naturally acts on $\mathcal{F}(X,K)$ by automorphism. Explicitly the action is $$h\cdot f :x\mapsto f(h^{-1}\cdot x)\text{.}$$

The wreath product of $K$ by $H$ is the semi-direct product of $H$ with $\mathcal{F}(X,K)$ using the action of $H$ by automorphism I have just described.

For your picture with the shoes, it looks like the idea. A more mathematical way to see a wreath product naturally appearing is when you are trying to describe centralizers of permutations in $S_n$ (the symmetric group over $n$ elements).

Example: Show that the centralizer of $(1,2)(3,4)(5,6)$ in $S_6$ is isomorphic to the wreath product of $\mathbb{Z}/2$ by $S_3$ (where $S_3$ naturally acts on $\{1,2,3\}$).

I give here some intermediate questions to compute the example. Denote $\sigma=(1,2)(3,4)(5,6)$ and $C$ the centralizer $\sigma$ in $S_6$.

1. Using the unicity in the theorem of decomposition of a permutation into a product of disjoint cycles, show that $C$ acts naturally on $\{\{1,2\},\{3,4\},\{5,6\}\}$.

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2. Show that the Kernel $K$ of this action is $\langle(1,2)\rangle\times\langle(3,4)\rangle\times\langle(5,6)\rangle$.

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3. Find a subgroup $S$ of $C$ of order $6$ such that $S$ acts faithfully, $3$-transitively on $\{\{1,2\},\{3,4\},\{5,6\}\}$.

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4. Conclude.

If you do the exercise, you will probably understand how to generalize this to describe the centralizer of any permutation in $S_n$ knowing its decomposition as a product of disjoint cycles (what you get is a direct product of wreath products).