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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Posted by Jason Polak on 28. January 2012 · Write a comment · Categories: books, modules · Tags: , ,

As it happens every so often, I browse the mathematical library pseudorandomly, and look out for interesting titles; usually a prerequisite for interesting is that they have something to do with the realm of algebra. This is exactly how I found Faith's book, with its captivating title urging me to borrow it.

Now, inevitably in mathematical research, one has to efficiently skim through papers and books to find specific ideas and facts. The unfortunate thing is that sometimes it is easy to neglect the stimulation of the idle curiosity that probably brought most mathematicians into their fields in the first place, and so I try to combat this neglect by my idle browsing and blogging.

I try not to spend too much time on this so that I progress with my degree, but I try to nurture my curiosity through reading anything that looks interesting. Returning to books, I do believe there are few worse literary follies than a graduate algebra textbook that lacks imagination in its examples and theorems and passion in its explication. I only fear that such books will tend to promote in the learning of higher algebra what most institutions have done with calculus, and that is to make it a tiresome mechanical effort, washing away the once vibrant and fanciful colours from the gentle tendrils of the mind.

But fear not! Should the mental dessication start to occur in a young algebraist's mind; should the flames of passion dim for the wonders of the injective module, she can always turn to the entire object of this post, videlicet Faith's "Rings and Things and a Fine Array of Twentieth Century Associative Algebra"
. I refer to the second edition, incidentally, which corrects many errors from the 1st edition.

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