The remainder of this chapter is devoted to the celebrated Sylow theorems, and this lecture contains the prerequisites. The Sylow theorems describe the structure of a finite group’s subgroups of prime power order, which are called p-subgroups. These theorems will allow us to prove a number of results about the structure of finite groups, including some that appear to be very unrelated. The proofs of these theorems involve clever group actions on sets of cosets and subgroups, and then the application the theory we’ve developed, such as the orbit-stabilizer theorem. We will also need to prove several results about the normalizers of p-subgroups, and that is what we will do in this video.
Course & book webpage (with complete lecture note slides, HW, exams, etc.): https://www.math.clemson.edu/~macaule/visualalgebra/
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0:00 Introduction
0:50 Cauchy's theorem
6:58 Classification of groups of order 6
10:36 Sylow theorem preview
12:41 The five groups of order 12 and their p-subgroup towers
13:21 The p-group lemma: |Fix(ϕ)| ≡ₚ |S|
15:49 Normalizer lemma, part 1: [N(H):H] ≡ₚ [G:H] for p-subgroups
17:34 Proof of the normalizer lemma, part 1
21:39 Picture: H acting on the right cosets of H
22:33 Normalizer lemma, part 2: normalizers of non-maximal p-subgroups grow
28:18 Coset partition interpretation of the normalizer lemma, part 2.
29:07 Proof of the normalizer lemma, part 2