Jordan-Holder

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!