Visual Algebra, Lecture 4.9: External products and holomorphs

Описание: In this lecture, we’ll formalize the construction of a semidirect product of two groups, algebraically. That is, we’ll pin down what the binary operation on the Cartesian product is. This is called an “external” semidirect product. An example of this is called the holomorph of a group, which is the semidirect product of a group with its automorphism group. We’ve actually seen a few examples of this, including almost a full construction in the previous lecture. We’ll finish with a preview of comparing "external" to “internal” semidirect products, which is when a group is isomorphic to a semidirect product of two of its subgroups.

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
1:00 Semidirect products, algebraically
4:49 Multiplication in the direct product C₅×C₄
7:10 Multiplication in the direct product C₅⋊C₄
9:28 A×{1} is normal in A⋊B
14:47 The holomorph of a group
17:49 The holomorph of C₅
19:28 The holomorph of V₄
20:55 The holomorph of D₃
21:48 Internal semidirect products
25:33 When is Hol(G)≅G×G?