Welcome back readers! In the last post, Dihedral Groups and Automorphisms, Part 1 we introduced the dihedral group. To briefly recap, the dihedral group $ D_n$ of order $ 2n$ for $ n\geq 3$ is the symmetry group of the regular Euclidean $ n$-gon. Any dihedral group is generated by a reflection and a certain rotation. Moreover, in Part 1 we gave two other descriptions of the dihedral group $ D_n$. The first is the presentation

$ \langle r, s | r^n, s^2, sr = r^{-1}s\rangle.$

We also discovered that if we consider the cyclic group $ C_n$ as a $ C_2 = \{ 1, \sigma\}$ module via $ \sigma*k = -k$, then $ D_n$ is isomorphic to a semidirect product: $ D_n\cong C_n\rtimes C_2$, which was the second description.

Now here in Part 2, we are going to learn something new about the dihedral group when $ n$ is even: in this case, $ D_n$ has an outer automorphism. But in order to prove this, we will introduce group cohomology!
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