How to Construct Every Symmetry | Decomposition, Extension, and Classification of Groups

Group is a mathematical abstraction used to describe symmetry. In this video, we introduce the theory of groups, group decomposition and extension, and computational aspects as to why classifying all finite groups is impossible.



Links:
Video that introduces sporadic groups
Why Do Sporadic Groups Exist?:    • Why Do Sporadic Groups Exist?  

Video that explains both algebraic and topological aspect of the spin group
Dirac's belt trick, Topology, and Spin ½ particles:    • Dirac's belt trick, Topology,  and Sp...  

Video for understanding the geometry of unit quaternions
Visualizing quaternions (4d numbers) with stereographic projection:    • Visualizing quaternions (4d numbers) ...  

Video that constructs the spin group as SU(2) axiomatically
The biggest misconception about spin 1/2:    • The biggest misconception about spin 1/2  

Entire video series on spinors
Spinors for Beginners:    • Spinors for Beginners  

Eilenberg and Maclane's original paper on group extensions with non-Abelian normal subgroup
Group Extensions with a non-Abelian Kernel: https://shorturl.at/ryFLZ

And here are some resources to learn more about group extension:
ncatlab: https://ncatlab.org/nlab/show/group+e...
groupprops: https://groupprops.subwiki.org/wiki/G...



Chapters:
00:00 Intro
01:21 Examples of Groups
02:50 Equivalent Groups
05:32 Abstract Definition of a Group
07:22 Definition of Isomorphism
08:24 Automorphism Group
10:28 Decomposition of Groups
14:48 Finite Simple Groups
17:07 Infinite Group Cannot Be Factored
18:33 Group Extension
19:18 Direct Product / Classification of Abelian Groups
24:14 Semidirect Product
26:51 Non-split Extensions
31:45 What is Cohomology
35:37 Why Classification of Groups is So Hard
36:32 Isomorphism Testing Algorithm
39:02 Status on Classification of Groups
39:46 Classification of Groups of Order 16
42:42 Asymptotic Group Theory
43:32 Chemistry Analogy
44:45 Outro


Music🎵:
No.3 Morning Folk Song – Esther Abrami    • No.3 Morning Folk Song  


Corrections:
0:37 60 = 2 × 2 × 3 × 5