Michi’s blog » archive for October, 2007

Wreath products

  • October 29th, 2007

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).

This is a preview of Wreath products. Read the full post (791 words, 19 images)

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Michi is a PhD student working in homological algebra. This blog is his outlet for texts with some manner of thought put into them. Over at his LiveJournal intimate details and streams of consciousness might be found.
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