Applications of the wreath product?

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?

Solution 1:

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.

Solution 2:

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 $C \times E$.

Solution 3:

(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.