Quipu

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Bibliographic Details
Online Access:	Full Text (via ProQuest)
Main Author:	Bae, Yongju
Other Authors:	Carter, J Scott, Kim, Byeorhi
Format:	eBook
Language:	English
Published:	Singapore : World Scientific Publishing Company, 2024.
Series:	K & E series on knots and everything.

Table of Contents:

Intro
Contents
About the Authors
Acknowledgments
1. Introduction
2. Background and Motivating Examples
2.1. Elements of group theory
2.1.1. Permutations
2.1.2. Permutation representations
2.1.3. Short exact sequences
2.2. (Z/2)2 and Z/4
2.3. (mod n)-quipu
2.3. (mod n)-quipu
2.3.1. Application: Z/n × Z/n and Z/(n2)
2.3.2. Fanciful application: The circle of fifths
2.3.3. Application: Dihedral groups
3. Matrix Descriptions
3.1. Overview
3.2. The 8-element dihedral group revisited
3.3. (mod 2)-quipu that describe a 24-element group
3.3.1. Actions upon the cube and an embedded tetrahedron
3.4. Signed permutation
3.4.1. Signs of permutations and determinant
3.4.2. The group and a matrix representation
3.4.3. Subgroups
3.4.4. The 4-dimensional cube
3.5. Semi-direct products
3.6. Proof of Theorem 1
3.6.1. An alternative view of the dihedral group D8
3.6.2. An alternative view of the alternating group A4
4. The 3-Dimensional Sphere is a Group
4.1. Balls and their boundaries
4.1.1. Boundaries of cubes
4.1.2. Boundaries of simplices
4.1.3. Taking the boundary twice
4.2. The unit vectors i, j, and k
4.2.1. The quaternions: Part 1
4.2.2. Stereograph projection
4.2.3. The quaternions: Part 2
4.2.4. Matrix representations
4.3. The dicyclic groups
4.3.1. Revisiting the dihedral group
4.3.2. Projection from S3 to the 3-dimensional rotation group
4.3.3. The dicyclic group as a subset of the 3-sphere
4.3.4. Matrices that correspond to the 2-strings-with-quipu representations
4.3.5. The dicyclic group Dic2 is the group of quaternions Q8
4.3.6. Describing the projections from the dicyclic groups to the dihedral groups
4.4. Culmination and anticipation
5. Extensions of the Permutation Group Σ4
5.1. The symmetric group Σ4
5.1.1. The cosets of the dihedral group
5.2. The group GL2(Z/3)
5.2.1. 4-strings-with-(mod 2)-quipu
5.2.2. The semi-dihedral group
5.2.3. A 3-strings representation of GL2(Z/3)
5.2.4. Peculiar correspondences
5.3. The group SL2(Z/4)
5.3.1. A 3-strings representation of SL2(Z/4)
5.3.2. The 2-Sylow subgroup of SL2(Z/4)
5.3.3. An alternative projection to Σ4
5.4. The binary octahedral group
5.4.1. Cosets of A = (1, a, a2,−1,−a,−a2)
5.4.2. Cosets of the dicyclic group Dic4 of order 16
5.4.3. Section summary
6. The Binary Tetrahedral Group
6.1. The binary tetrahedral group as a subgroup of S3
6.2. Correspondences among representations
6.2.1. Cosets of A = 〈a : a6 = 1〉
6.2.2. The subgroup Q8 of A4
7. The Binary Icosahedral Group
7.1. Powers of t
7.1.1. Cosets of T
7.2. Cosets of A
8. Computing Group 2-cocycles
8.1. Set-theoretic sections
8.1.1. Classifying seseqs
8.1.2. A geometric interpretation
8.2. Example 1. The symmetric group Σ4
8.3. Example 2. The group SL2(Z/4)

Quipu

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