Author Archives: tcgrubb

Hilbert’s Basis Theorem

If you have had some experience with rings and commutative algebra in the past, you may have encountered the idea of a Noetherian ring. This post will not deal with explaining Noetherian rings or their importance; instead I would recommend several … Continue reading

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“One Line” Algebra Qual Problems

Today I’m going to try to distill some solutions to previous algebra qual problems into one sentence answers. This will likely mean that the solutions are missing details, but my hope is to be able to give a decent overview … Continue reading

Posted in Group Theory, Ring and Field Theory | Tagged | Leave a comment

A good title for this blog post exists, but I don’t know what it is

Today we will look at one of my favorite topics in combinatorics: the probabilistic method. Championed by Paul Erdős in the 1940’s, the probabilistic method has since become a staple in extremal and Hungarian-style combinatorics. While the applications of the probabilistic method … Continue reading

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Groups of order 4,312 are not simple: An application of Sylow Theory

Today we will solve a problem from Wisconsin’s 2012 Algebra Qualifying Exam. The main tools used in the problem are Sylow Theory and group actions; we will take these tools for granted, but for great introductions to each topic I would … Continue reading

Posted in Group Theory, Uncategorized | Tagged | Leave a comment

Cauchy’s Theorem

The famous Lagrange’s Theorem states that, in any finite group , the order of any element divides the order of . A natural follow up to this theorem is whether or not the converse holds. That is, if divides the order of … Continue reading

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What are Permutation Patterns?

Today we examine a topic known as combinatorial pattern avoidance. There has been an explosion of interest in this field of late; see, for instance, the international permutation patterns conference, or any recent issues of your favorite discrete mathematics journal. I was … Continue reading

Posted in Combinatorics, Patterns | Tagged | Leave a comment

On the geometric mean of the binomial coefficients

Today we prove a remarkable asymptotic result regarding the binomial coefficients. Namely, let . We will show Before getting to the proof, I should say that this is most definitely not my creation; I believe the first publication of this … Continue reading

Posted in Asymptotics, Combinatorics, Contest Math | Leave a comment