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!?

Rectangles!

H

V

180

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

Matrices

**Group Theory**

-Why Bother?

Braid Structures

Permutations

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

Chemistry

Crystallography

X-ray diffraction

Organic Chemistry

Knot Theory

Topology

Cryptography

material science

Astrophysics

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

Quaternions

Construction of missing objects

String theory, particle physics - Standard Model

Understanding polarity, chirality,

optical isomers.