So when I got to grad school, certain gaps in my (very unconventional) math education revealed themselves, and one of them was that I didn’t know anything about vector space duality. I took a graduate linear algebra course that first semester, but I think what actually caught me up to speed was my peers. I …
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Gems from Isaacs’ book on characters iii: real and rational elements
This is part iii of the tear I’m on, writing down things I learned from I. Martin Isaacs’ book on characters of finite groups. Parts i and ii focused on controlling the degrees of irreducible characters, and I have more to share about that, but today let me focus on another theme: the group-theoretic conditions …
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Gems from Isaacs’ book on characters ii: more degrees
This is a direct followup of yesterday’s post. There is a bunch of great stuff I didn’t know in the exercises of Chapter 2 of Isaacs’ book, again much of it focused on getting control over the degrees of the characters of a finite group. Let me just get right into it: Proposition: Let be …
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Gems from Isaacs’ book on characters: controlling degrees
Something made me pick up I. Martin Isaacs’ book on character theory recently, and it is just full of great stuff. As a person who has a published paper in the representation theory of finite groups (journal | arXiv), maybe I expected I wouldn’t get much new info from the introductory chapters, but this is …
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Why nonmodularity makes the co/homology of a quotient so lovely
I had a conversation with Søren Galatius a while back in which he made use of the fact that for a particular topological space that was carrying an action by a particular finite group . When I wrote down my notes on the conversation, this point struck me as one of those “familiar, but why?” …
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Borel subgroups and Sylow subgroups
I’ve been reading T. A. Springer’s book on linear algebraic groups, and it’s really satisfying because it’s filling in a lot of details about things I’ve heard about for years but only in vague terms. More importantly, it’s filling in the story. I feel like I’ve looked up the definitions of the words “parabolic subgroup” …
Under the hood of the Steenrod 5-lemma
I first ran into the 5-lemma in an algebraic topology class. It was just tossed off with a comment like “as you can check” or something. Lemma (5-lemma): Suppose you have a commutative ladder diagram like so: If the rows are exact, and , , , and , are isomorphisms, then is an isomorphism. Since …
Burnside’s counting lemma via characters
I just noticed that there is a very conceptually transparent proof of Burnside’s counting lemma via character theory! Here’ s a formulation of the statement. (I have heard it is not really due to Burnside; Frobenius or something.) Burnside’s counting lemma: If a finite group acts on a finite set , the number of orbits …
Cohen’s theorem on noetherian rings
I recently inherited a copy of Nagata’s book Local Rings, and I’ve already learned a new theorem! Theorem of Cohen: A commutative ring is noetherian if and only if all its prime ideals are finitely generated. This is cool because if you, like me, have ever been sad about the fact that noetherianity is not …
Why x^p-a doesn’t factor unless it has a root
I’ve heard this result in the title referred to as “classical” but I’m actually not sure where to find the proof. It came up in conversation with a collaborator last week which is why I’m thinking about it. It’s a problem in Michael Artin’s Algebra text, somewhere in the Galois theory chapter, so I have …
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Every Single Problem 
