User:Jmracek

Hello all, and welcome to my user page. Here you will find notes, my assignment solutions, and assorted other bits that I find interesting.

11-1100 HW1 pt. 1 submission: MATLAB Code for Non-Commutative Gaussian Elimination

How to visualize normal subgroups

I was trying to find an intuitive way to just be able to instinctively tell whether or not a given subgroup is normal. I came across a few articles on the web that give some great insights, so I thought I would share them. I first came across an answer here:

http://forums.xkcd.com/viewtopic.php?f=17&t=26793

And then through reading the responses on the forum, was naturally led to an email exchange between John Baez and Sean Fitzpatrick:

http://math.ucr.edu/home/baez/normal.html

Let me summarize the main idea. Basically, one can tell whether or not a given subgroup is normal by finding other subgroups which, very roughly, "look the same". As an example, consider the permutations of vertices of the tetrahedron. One subgroup of order 3 can be pictured by rotating about an axis that goes through a vertex and the center of an opposing face; however, one could have equally well picked any of the other vertices of the tetrahedron to fix the axis. In fact, this subgroup is not normal.

The way I understand it is from the standpoint of linear algebra. Since we can picture the subgroup I mentioned as rigid body rotations in R^{3}, the most obvious representation is a subgroup of SO(3) that can be constructed by putting the center of mass of the tetrahedron at the origin, then finding the action of the group elements on an orthonormal basis. In this picture, conjugation by a group element can be roughly thought of as a coordinate change to some rotated frame of reference (because group conjugation is a similarity transformation in the matrix representations); so, if you can find a change of coordinates that makes the action of any two subgroups "look the same" then you know that it cannot be normal. In the previous example, one can see that any 4 of the subgroups of order 3 are essentially the same rotations, just looking down a different axis of the tetrahedron. I haven't worked it out in detail, but I think that the subgroup generated by <(234)> can be turned into the subgroup generated by <(123)> if you conjugate by (14).

More on how to visualize concepts in group theory

A very cool video explaining how to use Cayley graphs to visualize some group theoretic concepts. I especially enjoyed the part about the semidirect product.

http://web.bentley.edu/empl/c/ncarter/vgt/VisualizingGroupTheory-320x240.mov

Messages

Hey buddy. You might want to move your page so that that it has 11-1100 as the prefix. Tholden 14:32, 10 October 2011 (EDT)