Visual Algebra, Lecture 5.11: The first two Sylow theorems скачать в хорошем качестве

Visual Algebra, Lecture 5.11: The first two Sylow theorems

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... ------------------------------------------------------------------------------------------------------------------------------------------------------ 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