A maximal subgroup of prime power order is called a Sylow p-subgroup. In the previous lecture, we proved the first two Sylow theorem, and learned that all Sylow p-subgroups are conjugate. The third Sylow theorem says that the size of this conjugacy class is 1 mod p, and also divides the index m. This imposes strong restrictions on the structure of finite groups. In many cases, it is enough to show that there must be a unique Sylow subgroup, which means that it must be normal. In this lecture, we will prove this theorem, and use it to classify all groups of order 12. Then, we will see a number of ways to apply it to show that groups of a certain order cannot be simple, and a few other surprising results as well.
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
0:53 Summary of the Sylow theorems
2:56 The third Sylow theorem
7:32 Proof of the third Sylow theorem: showing there's a unique fixed point
13:37 Summary of the proofs of the Sylow theorems
15:04 Revisiting our mystery group of order 12
17:39 Classification of groups of order 12
24:50 There are no simple groups of order 84
29:48 There are no simple groups of order 351
34:42 There are no simple groups of order 24
40:10 The simple group GL₃(ℤ₂) of order 168
43:01 There are no simple groups of order 90