The most simple (and unhelpful) definition of group theory is as follows: group theory is the study of algebraic objects called groups. When mathematicians use the world algebraic they are referring to the branch of mathematics known as abstract algebra - this is the study of collections of objects that have some kind of "structure".
The integers
To illuminate this definition we can give an example. Consider the collection of both positive and negative whole numbers i.e. the integers. We normally represent these using the symbol $\mathbb{Z}$.
When we have two integers, let's call them $m$ and $n$, we can add them together to get a third integer $m+n$. We call addition a binary operation on $\mathbb{Z}$ as it takes two inputs and gives a single output. We may sometimes write this formally as $+ : \mathbb{Z} \times \mathbb{Z} \to \mathbb{Z}$.
This $+$ operation has a few interesting properties. The first property has to do with the integer $0$. For any integer $m$ (using mathematical shorthand we write this as 'for any $m \in \mathbb{Z}$') we have that $m+0 = m$. In other words adding $0$ to an integer simply gives us the same integer back. We call $0$ the identity on $\mathbb{Z}$.
For the second property take any $m \in \mathbb{Z}$. There is then an integer -$m \in \mathbb{Z}$ such that $m + -m = 0$. For example, if we take $m = 3$ we have -$m = $-$3$. Alternatively if $m = $-$4$ then -$m = 4$. We call -$m$ the inverse of $m$.
Finally, for the third property take any $m, n, s \in \mathbb{Z}$. We then have that
$m + (n + s) = (m + n) +s$.
This property is called associativity, and it may seem obvious to you at first! Consider however the subtraction operation - on $\mathbb{Z}$. Now it is not true that for any $m, n, s \in \mathbb{Z}$
$m - (n - s) = (m - n) - s$.
You can try to find actual numbers $m, n$ and $s$ so that the above equality doesn't hold if you are not convinced.
Definition of a group
This leads us on to the definition of a group. This should look quite familiar.
A group is a collection of objects $G$ together with a binary operation $\cdot : G \to G$ such that:
(1) There exists some $1_G \in G$ such that for all $g \in G$ we have $g \cdot 1_G = 1_G \cdot g = g$.
(2) For any $g \in G$ there exists some $g^{-1} \in G$ such that $g \cdot g^{-1} = 1_G$.
(3) For any $g, h, k \in G$ we have
$g \cdot (h \cdot k) = (g \cdot h) \cdot k$.
Although we have used slightly different notation ('$\cdot$' instead of '$+$', '$g^{-1}$' instead of '-$m$'), you can hopefully see that these are the exact same properties we showed for $\mathbb{Z}$. In fact it is true that $\mathbb{Z}$ is a group.
Note: we should more precisely write that ($\mathbb{Z}$, $+$) is a group as the definition of a group depends on both the collection of objects $G$ and the operation $\cdot$. Normally, however, the operation we are using is obvious so we just write $G$ instead of $(G, \cdot)$.
Other examples of groups
Many sets that you may have seen before can be given a group structure. Along with $(\mathbb{Z},+)$ above we can consider $(\mathbb{Q},+)$ and $(\mathbb{R},+)$. These are the sets of rational numbers and real numbers respectively. We have also the groups $(\mathbb{Q}^{\times},\cdot)$ and $(\mathbb{R}^{\times},\cdot)$, where the cross means we remove $0$ from the set. As an exercise see if you can work out why we need to remove $0$, and why $(\mathbb{Z}^{\times},\cdot)$ is not a group.
These are all examples of infinite groups. The area of group theory that I will be working in is called finite group theory. This is the study of finite groups. The most simple finite groups are called the cyclic groups. There is a cyclic group $C_n$ for every natural number $n$. We define $C_n$ as the group generated by some element $g$ of order $n$. Being generated by g means that $C_n = \{ g^m : m \in \mathbb{Z}\}$ i.e. an element of $C_n$ is of the form $g^m$, and $g$ having order n means that $g^n = 1$. As you might have guessed, $g^m \cdot g^r = g^{m+r}$.
There are many more complicated finite groups. For example, the groups I am studying are called the projective special linear groups over a finite field. I'll be talking about these in future posts.
Why do we care?
You may think that the properties we used to define a group seem quite arbitrary. Why should we care that the algebraic object we study has an identity, or inverses? This is a good point and in fact there are names for these objects that have less structure than groups. In my opinion, the reason that group theory is so widely studied is that these three properties give just enough structure to make groups interesting to think about while still being very simple to define.
The algebraic objects with less structure tend to be rather boring. The most extreme example is a magma, which is simply a set with any binary operation. We have so much freedom here that there is not much we can say about all magmas in general, so they are not widely studied.
The algebraic objects with more structure are more interesting, but tend to be harder to define. For example modules are incredibly useful throughout mathematics, but the definition of a module is much more involved than the definition of a group.
Groups also come up naturally in many areas of mathematics. Group theory was actually initially invented in the 18th century as a way of studying the roots of polynomials, and did not become its own area of study until later.
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