305 – A potpurri of groups

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.”

  • S_n, A_n, the symmetric and alternating groups in n letters.
  • Abelian groups, such as ({\mathbb Z}/n{\mathbb Z},+), (({\mathbb Z}/n{\mathbb Z})^*,\cdot),{\mathbb Z},{\mathbb Q}.
  • 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, GL_n({\mathbb R}), the group of all invertible n\times n matrices with real entries, or SL_n({\mathbb R}), the group of all n\times n 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.

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