Here are a few examples of groups and links illustrating some of them. I will be adding to this list; if you find additional links that may be useful or interesting, please let me know. A nice general place to look at is the page for the book “Visual group theory.”
- , the symmetric and alternating groups in letters.
- Abelian groups, such as .
- Dihedral groups. Here is a page by Erin Carmody illustrating the symmetries of the square. The Wikipedia page on dihedral groups has additional illustrations and interesting examples.
- Braid groups. Patrick Dehornoy has done extensive research on braid groups, and his page has many useful surveys and papers on the topic. Again, the Wikipedia page is a useful introduction. The applet we saw in class is here.
- Matrix groups. For example, , the group of all invertible matrices with real entries, or , the group of all matrices with real entries and determinant 1.
- The plane symmetry (or Wallpaper) groups.
- Coxeter groups.
- Crystallographic groups.
- Any group is (isomorphic to) a group of permutations, but the groups corresponding to permutation puzzles are naturally described this way. For example, Dana Ernst recently gave a talk on this topic.