Permutations

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.

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…