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Group Theory

A gentle introduction

Adam Haunch

on 4 February 2014

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Transcript of Group Theory

Modulo Arithmetic
Complete the table under addition mod 4.
Modulo Arithmetic 2
Complete the table under multiplication mod 5
Rotations of a Square
Complex Numbers
multiply the complex numbers and simplify
Group Theory
Or the fundamental connectedness
of all mathematics!!!
Algebraic Structures
These tables arise from a diverse set of mathematical areas, but some have the same structure!
Can you identify which!?
We have seen two distinct groups:
The Klein Group
The Cyclic Group
Bigger Groups!
Construct the group with elements {0,1,2,3,4,5} under addition mod 6.

Construct the group with elements{1,2,3,4,5,6} under multiplication mod 7
Isomorphic Groups?
There are six transformations on an equilateral triangle. Construct the group table.
Abstract and confusing groups
Infinite Groups!
The integers under addition
The rational numbers (without zero) under multiplication
The real numbers under addition
Complex numbers of modulus one
Group Theory
-Why Bother?
Braid Structures

What happens if we permute three objects?
Are either of these the same as the previous group?
String Theory
The Monster Group
The Leech Lattice
X-ray diffraction
Organic Chemistry
Knot Theory
material science
Shape of the universe
Cohomological constructs
Numer theory
Galois groups
Poincare Conjecture
Symmetries in physics
Comparision of Bosons and Fermions
The circle of fifths in music!
Analysis of fractals
deeper analysis of known objects
Platonic Solids
N,Z,Q numbers
real numbers
complex numbers
Extensions of known number systems
Construction of missing objects
String theory, particle physics - Standard Model
Understanding polarity, chirality,
optical isomers.
Full transcript