## 305 – A potpurri of groups

February 27, 2012

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.