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Michi’s blog » archive for October, 2007

 Wreath products

  • October 29th, 2007
  • 10:11 pm

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 H \wr_X G is defined for groups G,H and a G-set X by the following data. The elements of H \wr_X G are tuples (h_{x_1},h_{x_2},\dots,h_{x_r};g)\in H^{|X|}\times G. The trick is in the group product. We define
(h_{x_1},h_{x_2},\dots,h_{x_r};g)\cdot
(h’_{x_1},h’_{x_2},\dots,h’_{x_r};g’)= \\
(h_{x_1}h’_{gx_1},h_{x_2}h’_{gx_2},\dots,h_{x_r}h’_{gx_r};gg’)
(or possibly with a lot of inverses sprinkled into those indices)

Consider, first, the case of G=H=X=C_2 with the nontrivial G-action defined by gx=1, g1=x. We get 8 elements in the wreath product H \wr_X G. Thus, the group is one of the groups with 8 elements – C_8, C_4\times C_2, C_2^3, Q, D_4. 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).