Into The Möbius Labyrinth: Six Möbius Functions, One Nonabelian Machine | Group Theory | Dogmathic скачать в хорошем качестве

Into The Möbius Labyrinth: Six Möbius Functions, One Nonabelian Machine | Group Theory | Dogmathic

https://dogmathic.com/ Today we prove a finite set of Möbius transformations forms a group under function composition. We start with the domain A = R \ {0,1} so every function we use actually maps A to A. Then we define six functions: f(x) = 1/(1-x) g(x) = (x-1)/x h(x) = 1/x i(x) = x j(x) = 1-x k(x) = x/(x-1) From there we check the group axioms the clean way for a finite set: build the Cayley table for composition. The table instantly confirms closure, shows i(x)=x is the identity, and lets us read off inverses. Along the way I compute several compositions by hand so you can see exactly how these ugly fractions simplify. Once the table is complete, we extract structure: element orders (including f^3 = i and g^3 = i, while h^2 = j^2 = k^2 = i), a quick nonabelian check by comparing entries across the diagonal, and why associativity comes for free from composition. We finish with extra facts: these are also called fractional linear functions, this set is generated by two elements, we list subgroups, and we connect the whole structure to the familiar order 6 groups D3 and S3 via isomorphism.    • The Gateway to Group Theory: Groups in Und...      • Unmasking Cayley Tables: Why Z/5Z Breaks U...      • Abstract Algebra   PROPERTIES AND CONCEPTS USED Möbius Transformations And Fractional Linear Functions Domain Restriction A = R \ {0,1} Functions As Elements Of A Set Under Composition Group Axioms: Closure, Identity, Inverses, Associativity Cayley Table For A Finite Operation Identity Function i(x) = x Inverse Functions From Table Entries Composition Computations With Rational Expressions Element Order And Powers Under Composition Nonabelian Test Via Table Asymmetry Generating A Group From Two Elements Subgroups And Orders Dividing 6 Isomorphism To D3 And S3 Cross Ratio Group Naming CHAPTERS: 00:00 Introduction 00:55 Möbius Setup 02:05 Domain A 03:05 Six Functions 04:10 Group Checklist 05:10 Cayley Table 06:20 Identity Row 07:15 Compose f With f 08:40 Compose f With g And h 10:10 Compose f With k 11:35 Compose g With f 13:20 Finish The Table 17:10 Inverses And Squares 19:05 Element Orders 21:10 Associativity 23:30 Nonabelian Check 25:20 Names And Generators 27:10 Subgroups And Isomorphisms 29:00 Thanks For Watching #dogmathic #GroupTheory #MobiusTransformation #MoebiusTransformation #FunctionComposition #CayleyTable #AbstractAlgebra #NonAbelianGroup #ElementOrder #Subgroups #Isomorphism #D3 #S3