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Orbit-Stabilizer for Finite Group Representations One of my professors covered the main result of this post during a class that I missed awhile ago. Using some notes from a friend who attended that class, I want to try to reconstruct the theorem 1. Experience with representation theory will be useful for this post, but I’ll try to cover enough of the basics so that previous exposure isn’t strictly required. which, unsurprisingly, is a version of Orbit-stabilizer for representations of finite groups ↩ Groups Aren't Abstract Nonsense I’ve recently been skimming through this book called “Office Hours with a Geometric Group Theorist” which, perhaps unsurprisingly, is about using geometric objects to study groups. It mostly focuses on how group actions on graphs and metric spaces can reveal information about the group1, and contains some pretty nice results. Unfortunately, I have too many in mind for one post, but I would still like to introduce the basic notions of the subject and a few results I enjoyed. I imagine this wil... Difference of squares Two new posts in one day? It must be Christmas. I think this post will be relatively short. I want to talk about a problem that popped in my head while I was working on the last post, and then mention some thoughts that this problem sparked which I hope are worth writing down before I forget. An interesting equation One day I will return to writing posts that are not always very algebraic in nature, but this is not that day. I want to talk today about an example of a peculiar equation, but first a little background… In my mind, number theory (at least on the algebraic side) is ultimately about solving diophantine equations and not much more. This is what originally got me interested in the subject because trying to solve these equations can often feel like some sort of puzzle or exploratory game; there’s... Algebra Part II This is the second post in a series that serves as an introduction to abstract algebra. In the last one, we defined groups and certain sets endowed with a well-behaved operation. Here, we’ll look at rings which are what you get when your set has two operations defined on it, and we’ll see that much of the theory for groups has a natural analogue in ring theory.