TITLE: Applications of the wreath product?
QUESTION [36 upvotes]: 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?

REPLY [2 votes]: Here are two places the wreath product is used in "real" applications.

The representation theory of $S_m\wr S_n$ can be used to prove results about voting for committees of a certain type.  (Published at https://doi.org/10.1016/j.aam.2020.102077)
The wreath product has been used in analyzing a number of different music theoretic concerns; see Peck's article for a survey.

In both cases, basically the object of study and its symmetries forced consideration of the wreath product rather than a larger symmetric group.  I think that's pretty cool.