In this article we mostly talk about partial orders, and lattices, which are a special type of partial order. We tie it in with our discussions of subgroups before proving two distribution laws for lattices.

We’re continuing our proofs of Sylow’s Theorems, and also continuing our efforts to count orbits, and determine the size of orbits created by the permutation groups…

In the previous entry, we gave a laborious proof that a group with an even number of elements has to have an element of order 2. We can do much better than that. We’ll show how by proving some famous theorems that were originally due to Sylow.

We continue our discussion of groups and permutations, breaking down a permutation into disjoint cycles, finding a proper subgroup of even permutations in a group of order 2(2k + 1), and discussing properties of the orbits, stabilizers, fixes, etc…

Here we further explore some of the interconnections between groups and permutations on sets. This quickly leads us to some interesting results.

In the previous entry More Introductory Group Theory, we introduced the definition of a group and also provided some lemmas to help us understand their basic structure a little better. It wasn’t as thorough as what you’d find in most Abstract Algebra books, though. But we did manage to introduce various morphisms, or maps between groups that preserve parts of their structure. Here we’ll continue with that theme…

We’ve already introduced some of the topics of Group Theory. In this entry we’ll talk about Lagrange’s Theorem, various morphisms, and try to tie it back to our ongoing discussion of permutations. We’ll also give some examples of groups…

Here we prove that permutations can be divided into even permutations and odd permutations. We’ll also go over what happens when you compose two permutations. Surprisingly, it behaves like adding numbers. E.g. composing an even permutation with an odd permutation results in an odd permutation. Two odd permutations result in an even permutation, etc…

Here we show the classic result that the sum of 1/p, where p ranges over all prime numbers, is infinite.

The dot product is so fundamental to linear algebra that it’s easy to take for granted that it works. Here we take a quick look at some of it’s potential by showing how it can be used to calculate the angle between two vectors.