Examples of Alternating Groups (Algebra 1: Lecture 8 Video 3)

Lecture 8: In this lecture we focused on the sign of a permutation and the alternating group. We reviewed what we know about writing a permutation as a product of disjoint cycles, and writing a permutation as a product of transpositions. We stated an important theorem, that if we can write a permutation as a product of transpositions in two different ways, then those numbers of transpositions are either both even or both odd. This led to the definition of the sign of a permutation. We saw that sign gave a homomorphism from S_n to {+-1}. The alternating group is defined to be the subset of S_n consisting of even permutations, or equivalently, the kernel of the sign homomorphism. We worked with the example of A_4 and proved a statement we made about it earlier, that it has no subgroup of order 6. In the last video, we saw two additional perspectives on the sign of a permutation, first in terms of permutation matrices and determinants, and then in terms of what a permutation does to a special polynomial Prod (x_i-x_j).

Reading: The main reference for this lecture is Conrad's notes 'The Sign of a Permutation'. I followed Section 1 and 2 of these notes pretty closely in the first two videos. In the last video, I followed Section 3 on permutation matrices pretty closely. The final part of the last video is a summary of the material of Section 3.5 of Dummit and Foote. In the third video, I gave the first proof that A_4 has no subgroup of index 2 given in Conrad's notes: 'No Subgroup of A_4 has Index 2'. There are several other proofs of this fact given in these notes-- I recommend that you read them all.