Introduction to Groups

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Abstract algebra can be broken into about two or three sections: (1) Groups; (2) Rings and Fields; (3) Vector Spaces. (A fourth topic that could be considered its own is Galois Theory.) The typical way this version of algebra is introduced is to start by covering groups, then to introduce the concepts of rings and fields in the context of group theory. Rings are interesting, but they are not of interest to us right now. In this post, I attempt to define what a group is and to explain what the definition means. I then follow this up with examples of groups using the real numbers, integers, and complex numbers. In the next post, I will attempt to prove that the real numbers and integers are groups under addition and that the complex numbers form a group under complex multiplication. To that end we must introduce the concept of a group.

Definition. A group $G$ is a non-empty set equipped with a single binary operation $\ast$ that satisfies the following four axioms:

For all $g_{1}, g_{2}\in G$ , $g_{1}\ast g_{2}\in G$ ;
For all $g_{1}, g_{2}, g_{3}\in G$ , $g_{1}\ast (g_{2}\ast g_{3}) = (g_{1}\ast g_{2})\ast g_{3}$ ;
For all $g\in G$ , there exists an element $g^{-1}\in G : g \ast g^{-1}=e_{G}=g^{-1} \ast g$ ;
For all $g\in G$ , there exists an element $e_{G}\in G$ such that $g \ast e_{g}=g=e_{g} \ast g$

This definition might not seem very helpful, but it’s the reason as to why we are allowed to use the properties of numbers that we used in high school. What this definition says is that we have a collection of objects that we call elements that is endowed with a binary operation. An example of this would be traditional addition or multiplication. The word binary simply means that it requires two elements to produce another element. For instance, consider the following group $G=(\mathbb{Z},+)$ (we’ll discuss what this notation means later on in the post). Let $a=2\in \mathbb{Z}$ and let $b=3\in \mathbb{Z}$ . We can add these two integers to get another integer, call it $c=5\in \mathbb{Z}$ . This is what we mean by a binary operation.

Now that we understand what the operation is, we can get into what it takes for a set to be a group. Statement 1 simply says that given any two elements in the set $a,b$ , if $a \ast b$ is also in the set, then we say that the set is closed under the operation $\ast$ . Statement 2 says that given any three elements in the set, the order in which the operation is performed, as dictated by the parentheses, is immaterial. This statement ensures that the elements of the set are associative. To clarify this a bit more, consider the following sum in $(\mathbb{Z},+)$ :

$2 + (3 + 4)$

The second statement says that I can remove the parentheses from the sum without it changing value. Indeed, in each case one gets 9, and so that’s what statement 2 ensures: associativity. Statement 3 says that there is an inverse element to each element in the set. For addition, this inverse element is the additive inverse, by which I mean that for every $a\in G, a+(-a)=e_{G}$ . For multiplication, the inverse element is the multiplicative inverse, in which case we have that for every element $a\in G, a\cdot a^{-1}=e_{G}$ . In the definition, I was careful to include both the right and left inverse. The reason for this is not all sets of elements commute, that is it is not always true that $ab=ba$ . As an example, let $M(2,\mathbb{R})$ be the set of all $2\times 2$ matrices whose determinant is non-zero. Matrices are known to non-commutative, and so $\textbf{AB}\neq\textbf{BA}$ . If it is true that every element in the group commutes, then $G$ is referred to as an abelian group.

Finally, statement 4 ensures that there is an identity element. In the definition and explanation of statement 3, I denoted this as $e_{G}$ . For addition, the identity element is $0$ , since given any element of $G$ , if we add $0$ to it we get the element again. Furthermore, for multiplication, the identity is 1 since anything multiplied by 1 is itself. Note that for addition and multiplication we have different forms for the inverse and identity elements. Thus, we can write these sets in one of two ways: in additive notation $a+b$ , and in multplicative notation $ab$ , depending on the operation involved. If all four of these axioms, as we call them, are satisfied, then the set is a group under the prescribed operation.

The typical examples of groups include the real numbers $\mathbb{R}$ , the integers $\mathbb{Z}$ , and the complex numbers $\mathbb{C}$ to name a few. We denote the structure by stating what set we are considering, followed by a the binary operation, written as $(G, \ast)$ .

In the next post, I will look at $\mathbb{R}, \mathbb{Z}, \mathbb{C}$ in detail and I will show how to prove that the reals and integers are groups under addition and that the complex numbers are a group under complex multiplication.

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