Category Archives: math

Anything mathematical.

Computing the Alexander polynomial: a guide

Given a knot $K$, which is an embedding $S^1\to \R^3$, we have see how to compute the fundamental group of $K$, defined as $\pi_1(\R^3 – K)$. For example, we have computed the fundamental group of the trefoil knot and the fundamental group of the cinquefoil knot. The fundamental group of the trefoil can be given […]

The fundamental group of the cinquefoil knot

It’s knot learning time! Last time, we looked at the trefoil knot: In that post, I showed you how you can write down a presentation for the fundamental group of a knot. So, you may want to review the procedure I gave in that post, because we’re going to do it again! This time, I […]

Fundamental groups: trefoil knot is not the unknot

Let’s learn about knots! I’m sure we’ve all experienced knots. For example, if you have a bunch of computer cables, chances are they have knots in them. Knots are actually quite tricky. Of course, first, we should say what a knot is, should we not? A knot is a continuous map $S^1\to \R^3$ which is […]

A short tutorial on Tietze transformations

Here is a little tutorial on how to use Tietze transformations. They were named after Austrian mathematician Heinrich Franz Friedrich Tietze. A presentation is a set of generators and relations given by the notation $$\langle~ S~|~W~\rangle$$ where $S$ is set and $W$ is a set of words in the symbols of $S$. $S$ is called […]

List: groups up to order 15 with proofs

One of the prime goals after any mathematical structure is defined is to classify all possible structures up to isomorphism. Here, we will do something a little more modest: classify all finite groups up to order fifteen. Despite spending a great amount of time with groups, I’ve never actually done this formally, although I’ve certainly […]

Is Aut(GxH) = Aut(G)xAut(H) for groups G, H?

Let $G$ be a group. The automorphism group of $G$ is the group of all isomorphisms $G\to G$. We denote this group by ${\rm Aut}(G)$. Is it necessarily true that ${\rm Aut}(G\times H) \cong {\rm Aut}(G)\times {\rm Aut}(H)$? Suppose $G$ and $H$ are finite groups whose orders are coprime. Then any homomorphism $f:G\to H$ must […]

The nonvanishing K_2(Z/4)

We saw previously that $K_2(F) = 0$ for a finite field $F$, where $K_2$ is the second $K$-group of $F$. It may be helpful to refer to that post for the definitions of this functor. I thought that it might be disappointing because we did all that work to compute the second $K$-group of a […]

The K-theory of finite fields: a synopsis

In my previous post, I proved that if $F$ is a finite field, then multiplicative group $F^\times$ is a cyclic group. This fact will play a small part in our description today of the $K$-theory of $F$. We will start by describing the classical $K$-theory of $F$ and then briefly talk about Quillen’s computation of […]

The multiplicative group of a finite field is cyclic

Today, we are going to prove that if $F$ is a finite field, the the multiplicative group $F^\times$ of $F$ is a cyclic group. Let’s start with an example of this phenomenon. Take $\F_5 = \{0,1,2,3,4\}$. Now, if we take powers of $2$ modulo $5$ we get $$2,2^2=4,2^3 = 3, 2^4 = 1.$$ Therefore, we […]

Check out my Sudoku solver

A couple days ago I started solving a few Sudokus. I would say I am average at solving them. I certainly don’t use any advanced techniques and I don’t really find the difficult ones very entertaining. By difficult here I mean where you have to think several moves ahead to rule out possibilities. So I […]