Groups Lecture 1

in which we meet the group axioms.

In our first lecture we explored what the examples of adding mod 3 and composing symmetries of a regular triangle have in common. This lead us to define a more abstract concept based on these common properties:

A group is a set $G$ with an operation $*$ satisfying
0. Closure : for all $a,b\in G$ we have $a*b\in G$.
1. Identity: There is $e\in G$ such that for all $a\in G$, we have $a*e=a=e*a$.
2. Inverses: For each $a\in G$ there is $a^{-1}\in G$ such that $a*a^{-1}=e=a^{-1}*a$.
3. Associativity: For all $a, b, c \in G$, we have $a*(b*c)=(a*b)*c$.

If the group also satisfies
4. Commutativity: For all $a,b\in G$, we have $a*b=b*a$,
then the group is called abelian.

We looked at several examples and counterexamples, and you can find more such to do yourself on the Practice Sheet A. Remember in particular the example of a non-associative operation, such as “to the power of”: $(2^3)^2=8=64\neq 512 = 2^9=2^{(3^2)}$.

We also started to look at some simple properties of groups, but we’ll come back to that in the next lecture.

Understanding today’s lecture

You will have to get used to using the axioms, so I strongly suggest you go through some (or even all!) of the exercises on Practice Sheet A. Also play around a little with symmetries of a regular triangle (or a square…) to get a feel for associativity and when things commute or not. If it’s all too easy for you (perhaps you’ve met groups before), then try the last question on Sheet A.

Going a little deeper

We said fairly casually that an operation is a way of combining two elements to get a new element, for example addition of integers, composition of symmetries, etc. The formal definition is this:

An operation on a set $X$ is a function $*\colon {X\times X\to X}$.

This means that we take an ordered pair $(a,b)$ of elements in the cartesian product $X\times X=\{(a,b)\mid a,b\in X\}$ and send that ordered pair to another element of $X$: $*(a,b)=a*b\in X$. This is the reason we have given the closure axiom the number 0: it is really already contained in the fact that $*$ is an operation on the set $G$, but we write it out explicitly so we don’t forget to check it when showing that something is a group.

Preparing for Lecture 2:

The idea of making an abstract definition is that we can prove results just using these properties, rather than proving them for each example separately. We will continue exploring this next time. For instance, in the examples we’ve seen, we always have a unique identity element. We proved just using the axioms (in fact, just using the identity axiom) that this is true in all groups. Can you work out how to prove the second statement, that inverses are unique? Or work out what happens if you want to take the inverse of a product?