In a conversation on IRC, I started prodding at low-order wreath products. It turned out to be quite a lot of fun doing it, so I thought I’d try to expand it into a blog post.

First off, we’ll start with a **definition**:

The wreath product is defined for groups G,H and a G-set X by the following data. The elements of are tuples . The trick is in the group product. We define

(or possibly with a lot of inverses sprinkled into those indices)

Consider, first, the case of with the nontrivial G-action defined by gx=1, g1=x. We get 8 elements in the wreath product . Thus, the group is one of the groups with 8 elements – . We shall try to identify the group in question using orders of elements as the primary way of recognizing things. Consider an element ((x,y),z).

Wreath products. Read the full post (791 words, 19 images)