Visual Algebra, Lecture 2.10: Examples of semidirect products

Описание: We begin this lectures by construct two nonabelian semidirect products of C_5 with C_4 with our "inflation method," where rewired Cayley graphs are inserted into inflated nodes of the other Cayley graph. We then analyze which groups we get. Then, we revisit four groups of order 16 that we have seen recently, which are all semidirect products of C_8 with C_2. After that, we construct the smallest nonabelian group of odd order, a semidirect product of C_7 with C_3. We finish with a surprising fact, of how the dihedral group D_n is not only a semidirect product, but also a direct product, when n is even.

Course & book webpage (with complete lecture note slides, HW, exams, etc.): https://www.math.clemson.edu/~macaule...

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CHAPTERS

0:00 Introduction
1:16 Recap of the "inflation method"
3:03 Recap of the rewiring of C_5
4:52 The first semidirect product of C_5 with C_4
6:39 The second semidirect product of C_5 with C_4
7:59 The two semidirect products, side-by-side
10:18 The second semidirect product of C_5 with C_4
12:12 Questions about our semidirect product construction
18:03 Which groups of order 20 did we just construct?
26:46 Four familiar groups of order 16
27:34 Constructing a semidirect product of C_8 with C_2
29:32 The four semidirect products of C_8 with C_2
34:12 Semidirect products of C_{2^m} with C_2
37:52 The smallest nonabelian group of odd order
42:17 Dihedral groups as direct products