In Linear Algebraic Groups 2, we defined the Lie algebra of an algebraic group $ G$ to be the Lie algebra of all left-invariant derivations $ D:k[G]\to k[G]$ where $ k[G]$ is the representing algebra of $ G$. However, we were left trying to figure out exactly how a morphism $ \varphi:G\to H$ determines a morphism $ d\varphi:\mathcal{L}(G)\to\mathcal{L}(H)$. It turns out that the answer is slightly tricky, and thus the Lie algebra in terms of left-invariant derivations $ k[G]\to k[G]$ really can't be the "real" or "most natural" definition!

Now, I could just give the formula for morphisms right away, but I think it would be a bit unmotivated. So before looking at the formula, let us first look at another way of defining the Lie algebra of an algebraic group. In fact, the next defintion is much more natural in that the functoriality of the Lie algebra construction will be clear, whereas in this case it was not.
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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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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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