Sets

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…

A permutation is a function from a set into itself that is both 1 - 1 and onto. You probably know what those terms mean already, but just to refresh, a function is said to be one to one (1-1) if f(x) = f(y) implies x = y.