In this entry we prove that A_n is simple when n >= 5. We then use this result to prove that there is no generalization of “even” and “odd” permutation to three or more classes – i.e. there is no class of homomorphisms into Z_n, where n > 2.
This entry starts out by building to Zassenhaus’ Lemma, and finishes by proving the Jordan-Holder theorem about subnormal sequences for finite groups. Enjoy!
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…
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…
