# The definition of a group

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## Introduction

In the [previous session](https://hackmd.io/@alexhkurz/SJZTQ9moL) we found 4 different ways of doing "addition" on the 4 numbers $0,1,2,3$.

The first operation was the following.

0 1 2 3
1 2 3 0
2 3 0 1
3 0 1 2

This one is indeed called addition, or, more precisely, addition modulo 4. We see that it is reasonably similar to the familiar addition of numbers we know. After all, we can't have $1+3=4$ because there is no number 4.

**Activity:** Explain why to define $1+3=0$ is the "next best thin" one can do if $1+3=4$ is not available? Can you find real world examples where this addition is used?

Then we also had the following 3 tables.

0 1 2 3
1 0 3 2
2 3 0 1
3 2 1 0

0 1 2 3
1 0 3 2
2 3 1 0
3 2 0 1

0 1 2 3
1 3 0 2
2 0 3 1
3 2 1 0

In this session we disucss the following question.

**Question:** What do the 4 tables have in common? Which laws of computation do they all satisfy?

## Definition of a group

As these laws of computation are so important, mathematicians have given them a name. Let us put this into a definition.

**Definition:** A ***group*** is a set of elements with a distinguished element $e$, called the identity, or unit, or neutral element. Moreover, for any two elements $x,y$ of the group there is an element $x\cdot y$. This operation is called the multiplication of the group. Multiplication must be *associative*, that is,
$$ x\cdot(y\cdot z) = (x\cdot y)\cdot z$$

for all elements $x,y,z$ of the group. The neutral element must satisfy the law of identity, namely,
$$ e\cdot x= x\cdot e =x$$

for all elements $x$ in the group. Furthermore, for every element $x$ of the group there is an *inverse* $x^{-1}$ satisfying
$$ x\cdot x^{-1} = e = x^{-1}\cdot x$$

Finally, the group is called ***abelian*** if it is commutative, that is, if
$$ x\cdot y = y\cdot x$$

for all elements $x,y$.

## Postscript

**Hint for the Activity:** What does the clock show when you add 1 quarter of an hour to 3 quarters of an hour?