## Dihedral Groups and Automorphisms, Part 1

Posted by Jason Polak on 22. March 2012 · 1 comment · Categories: group-theory · Tags: , ,

Welcome to the second two-post series on AZC! The evil secret plot of this series is to make group cohomology seem interesting for those who have not seen much group cohomology. To do this, we will dissect the dihedral group, which most math majors have probably seen as undergrads. However, to be thorough, the first post (i.e. this one) will described the dihedral group, and only in the second (will be posted soon) will we bring in group cohomology.

The finite dihedral groups are a good, concrete example of finite groups because they are not abelian and yet are not too convoluted for a blog post. There are many ways to define the dihedral groups, but the one that perhaps gives the most context and motivation is the definition in terms of symmetry of equilateral polygons in the Euclidean plane.

For an integer $n \geq 3$, the dihedral group $D_n$ is the symmetry group of rotations and reflections of the Euclidean planar regular $n$-gon. Let us look at the case $n = 3$ in a bit more detail. By labelling the vertices of an equilateral triangle with $\{a,b,c\}$, we can deduce that $D_3$ has order six. In fact, any fixed letter can be rotated to any given position, and then the remaining two letters can be permuted via a reflection, so all of the $3! = 6$ possible configurations obtainable with reflections and rotations.
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