The three Sylow theorems tell us about the structure of a group's subgroups of prime-power order, which are called. p-subgroups. The 1st Sylow theorem tells us that p-subgroups of all possible orders exists, and they are nested: each non-maximal one is contained in a bigger one. In other words, they come in "towers" in the subgroup lattice. The 2nd Sylow theorem says that the maximal p-subgroups---the "tops of these towers"---form a conjugacy class. This means that there is a unique p-group tower for each prime dividing the order of G. The 3rd Sylow theorem imposes strong restrictions on the size of this conjugacy class, but we’ll save that, and its applications to simple groups, for the next lecture. Throughout this lecture and the next, we'll revisit an unknown group of order 12, and deduce as much as we can about its structure.
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
0:56 Goals of the Sylow theorems
2:03 The five groups of order 12
3:16 Notational conventions
5:21 Our unknown group of order 12
6:38 The first Sylow theorem
14:34 Revisiting our unknown group of order 12
15:02 The second Sylow theorem
16:37 Proof of the strong second Sylow theorem
23:57 14:34 Revisiting our unknown group of order 12
24:25 Sylow subgroups of the alternating group A₅
30:21 The normalizer of the normalizer