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We recently went through the wreath product in my group theory class, but the definition still seems a bit unmotivated to me. The two reasons I can see for it are 1) it allows us to construct new groups, and 2) we can use it to reconstruct imprimitive group actions. Are there any applications of the wreath product outside of pure group theory?

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I have seen the wreath product used by composers of "New Music." –  charles.y.zheng Mar 18 '11 at 15:46

7 Answers 7

I don't have much knowledge about Automata theory, but here is a paper, which presents, application of Wreath Products to Rational Languages.

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Even within Group Theory, wreath products have more interests than you note; I'll give a couple below. But to answer your question, though perhaps not very satisfactorily, you can define wreath products of semigroups in precisely the analogous way as you do for groups. Semigroups are closely related to (and key to understanding) automata theory (which itself has many applications), and wreath products can play an important role in the study and construction of come automata.

I say it may not be very satisfying, because it sounds as if I'm saying "Sure! It has lots of applications in "pure semi group theory!"...

But within Group Theory, one very important property of wreath products is the theorem of Kaloujnine and Krasner:

Theorem. Let $H$ and $K$ be any groups. If $G$ is an extension of $H$ by $K$ (that is, $G$ contains a normal subgroup $N$ such that $N\cong H$ and $G/N\cong K$), then $G$ is isomorphic to a subgroup of the wreath product $H\wr K$. That is, $H\wr K$ contains isomorphic copies of every extension of $H$ by $K$.

In principle, if you understand all simple groups and you understand all possible extensions of two given groups (in terms of the groups, perhaps), then you understand all finite groups. Though this "in principle" is hopeless in practice, it can be useful in specific circumstances.

For instance, the iterated wreath product of $\mathbf{Z}_p$ with itself plays an important role in the study of $p$-groups, and is associated to the $p$-Sylow subgroups of symmetric groups.

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Yes, the Krasner–Kaloujnine Theorem is very important. +1. –  HJRW Mar 18 '11 at 18:52

It's not exactly an application, but the Rubik Cube group provides some insight into the reason why wreath products are interesting and natural objects to study.

The cube has 8 corner cubies each with three faces, and 12 edge corner cubies each with 2 faces. If you imagine all permutations of the faces of the corner cubies that permute the cubies and may also rotate them through 120 or 240 degrees, then you get a group $C$ which is the (permutation) wreath product of a cyclic group of order 3 and the symmetric group $S_8$, and $|C| = 3^88!$. Simiarly the group $E$ of permutations of the faces of the edge cubies that permute the cubies and may also flip them through 180 degrees is the wreath product of a cyclic group of order 2 and $S_{12}$ and has order $2^{12}12!$.

The Rubik Cube group itself is a subgroup $G$ of the direct product $C \times E$. It turns out that only $1/12$ of the possible permutations in $C \times E$ are attainable without taking the cube to pieces and reconstructing it, so $G$ has index 12 in $C \times E$.

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For some strange reason this was the first example of a wreath product that occurred to me, too! Inside the Rubik Cube group we only have as a subgroup a copy of $A_8$ acting on the zero sum subspace of $C_3^8$ and as another subgroup a copy of $A_{12}$ acting on the zero sum subspace of $C_2^{12}$. Their direct product $H$ is of index 2 in the whole group. None of the order 4 generators are in $H$ as their restrictions to either edges ore corners is an odd permutation. –  Jyrki Lahtonen Jul 19 '11 at 17:51

The Lamplighter group is a nice group constructed via the wreath product. It is an example of a group of exponential growth which is still amenable and the notion of amenability is not pure group theory anymore.

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This lengthy comment expanding on your answer was a suggested edit by an anonymous user. I quote with some added formatting: "Lamplighter groups serve as an example for many interesting phenomena, for example: $\mathbb{Z}^3 \wr \mathbb{Z}_2$ is an example of a amenable group which is not Liouville (admits non-constant harmonic bounded functions). Many Lamplighter groups have been proved to have pure point spectrum. Lamplighter groups provided the first examples of random walks on groups whose distance from the origin grows sublinearly (slower than $c \cdot n$) but faster than $\sqrt{n}$. –  t.b. Jul 19 '11 at 16:39
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($\sqrt{n}$ is called "diffusive" behaviour - such as the case of random walks on $\mathbb{Z}^d$, while linear distancing happens e.g. for random walks on the free group, or more generally for any non-Liouville Cayley graph)" –  t.b. Jul 19 '11 at 16:40

(Iterated) wreath products have been recently used to construct a new kind of Galois representation. An arboreal Galois representation is a continuous homomorphism from the absolute Galois group of a field to the automorphism group of a rooted tree. Such representations occur naturally in arithmetic dynamics, when one starts with a fixed polynomial or rational function and considers its iterates under composition.

This paper of Boston and Jones gives a nice introduction to this subject.

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I just came across this and I wanted to add that One-relator groups and One-relator products have been studied extensively and still many open conjectures left. More generally, one tries to find solution to an equation in an over-group. Here is a thoerem involving Wreath product due to Levin:

T H E O R E M . Let $G$ be an arbitrary group, $C = gp(c)$ be a cyclic group of order $n$. A solution of equation $xa_0xa_1 ... xa_{n-i}= 1$, $a_i\in G$ is given by $c^{-1}f \in G Wr C$, where $f(c^i)=a_i^{-1}$ , $i= 1,...,n-1.$

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Are you saying that one-relator groups and products encode equations? (I think you need to make your post clearer...) –  user1729 Sep 13 '13 at 14:04
    
The user was asking for applications of wreath product. One of the problems in algebra is solving equations over group G. This is not possible in general. The above theorem tells you a particular case when it can be solved. So yes, one-relator groups and products encode equations. –  Hesky Cee Sep 18 '13 at 10:45

Consider the permutational wreath product to prove two classical theorems in groups theory.

First, given an action $G\times R\to R$ we mean a map $(g,r)\mapsto\ ^gr$ which satisfy:

a) $^1r=r$

b) $^{gh}r=\ ^g(^hr)$,

c) $^g(rs)=\ ^gr ^gs$.

So we can define $R\rtimes G$ as the group with undelying set $R\times G$ and the operation $$(r,g)(s,h)=(r\ ^gs,gh).$$ The $R\rtimes G$ is called the semidirect product of $R,G$.

Now assume that there is a left action $\Sigma\times G\to\Sigma$ specified by $(x,g)\mapsto xg$.

Let $A$ be a group, and let us denote with $A^{\Sigma}$ the set $\{f:\Sigma\to A\}$ that with operation $f_1f_2(x)=f_1(x)f_2(x)$, the set $A^{\Sigma}$ is group naturally.

So we can have an action $$G\times A^{\Sigma}\to A^{\Sigma},$$ by $$^gf(x)=f(xg).$$

With this, the permutational wreath product is defined by $$A\wr G=A^{\Sigma}\rtimes G.$$

This is exploited to prove the Nielsen - Schreier and Kurosh's subgroups theorems by employing the functorial properties of $\wr$,

To prove the Nielsen - Schreier's theorem, for example, one is conduced to consider the universal property definition for free groups, then compelling us to seek a diagram: enter image description here

to find a unique $\overline{\alpha}$ for each $\alpha$ and each $G$ which makes the diagram commutative: i.e. $\alpha=\overline{\alpha}\circ i$, where $B$ is a set of transversals of $F$ module $H$.

The solution is encoded taking in consideration a diagram which look like:

enter image description here

where naturally $$\overline{\alpha}=\pi_G\circ\alpha\!\wr\!\rho H\circ \varphi|_U,$$ is the desired extension, and where $\Sigma$ are the right cosets $H\setminus G$ and $\rho: F\to S_{\Sigma}$ is an associated permutation representation of $F$ in the symmetric group of $\Sigma$.

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The reference is Ribes and Steinberg's "A wreath product approach to classical subgroup theorems". Enseign. Math. (2) 56 (2010), no. 1-2, 49–72. –  janmarqz Feb 2 at 3:25

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