Applications of Automorphisms to Classifying Groups (Algebra 1: Lecture 13 Video 4)

Lecture 13: We started this lecture by proving that A_n is simple for all n \ge 5. We first needed to prove a lemma that said that any nonidentity element possesses a conjugate with a particular property. For the rest of the lecture we focused on Automorphisms. We first saw how the action of G by conjugation on a normal subgroup H led to a subgroup of the group of automorphisms of H. We then saw a bunch of applications. We defined the subgroup of inner automorphisms of a group (the ones that come from conjugation). We looked at some examples and asked when Inn(G) was equal to all of Aut(G). At the end of the lecture, we saw some applications of these ideas to classifying groups of given order. For these applications we briefly talked about GL(n,p).

Reading: The argument we gave for the simplicity of A_n came directly from Conrad's notes. (See page 4 starting with Lemma 3.2.) This is a version of the proof given by Dummit and Foote in Section 4.6. Section 4.4 of Dummit and Foote is all about automorphisms. We followed the presentation very closely. The statements about GL(n,p) don't come until later in the text, but we decided to include them here.