An Intro to Higher Mathematics via Group Theory

The transition from grade school mathematics to a mathematics specialist program in university is a big one, even for those who are in enriched high school mathematics programs. The primary change in my opinion is the shift from computations to abstractions. In grade school mathematics, the emphasis is on concrete examples involving numbers. In university mathematics and beyond, the emphasis is on abstract objects generalizing the concrete objects dealt with in earlier grades.

My favourite example of the above phenomenon is the concept of a group. This concept is usually introduced in the second or third year of a mathematics specialist program. It generalizes concepts encountered in earlier grades such as integers, rational numbers, complex numbers, geometric transformations and functions.

What is a group? Basically, it is a set $G$ with a binary operation taking pairs in $G$ to another element in $G$ satisfying basic properties we are familiar with. The formal definition is as follows:

A group $G$ is a set with a binary operation, $\bullet$ , such that the following conditions are true:

a) Closure: For any elements $g$ and $h$ , $g\bullet h$ is in $G$ .

b) Identity: There is an element $e$ in $G$ such that $g\bullet e = e \bullet g = g$ .

c) Associativity: For all elements $f$ , $g$ and $h$ in $G$ , $(f\bullet g) \bullet h = f\bullet (g \bullet h)$ .

d) Inverse: For all $g$ in $G$ , there is an element $g^{-1}$ such that $g\bullet g^{-1}=g^{-1} \bullet g=e$ .

Before we discuss implications of these axioms in a subsequent post, note that many of the concepts introduced in grade school mathematics are groups. For example, integers form a group where $\bullet$ is $+$ . As well, non-zero rational numbers also form a group where $\bullet$ is $\times$ . Functions form a group with $\bullet$ being pointwise addition. A slightly more subtle example is the group of rigid transformations of a regular $n$ -gon (eg. equilateral triangle, square, hexagon, octagon) consisting of rotations about the centre and reflections across diagonals. This latter group is called the dihedral group.

Hence, we see that groups generalize many of the concepts taught earlier in grade school under one roof. By deducing properties of groups based on the above axioms, we can say much in common about concepts such as integers, rational numbers and transformations.