Math Love Stories: Season 1, Episode 2 — The Context Behind the Definition of a Group

In our inaugural episode, we explored the broad question of “What is Abstract Algebra?” We were introduced to significant elements such as equations, symmetries, permutations, and _algebraic structures_.

> My objective in this series resembles an “art appreciation” approach. My lofty ambition is to unveil the beauty of advanced mathematics to everyone. If that sparks your interest, continue reading!

Deconstructing/Abstracting

In this episode, we will break down the process of solving a straightforward equation, which will lead us to one of the core definitions in abstract algebra.

Consider the equation:

What value of x satisfies this equation?

We don’t require advanced algebra to solve this. It’s universally agreed that one plus two equals three, thus x = 1 is the solution.

You might think starting with such a basic equation is trivial, but there's a deeper significance beneath the surface.

Mathematics often presents us with challenging problems, and a favored technique is to simplify these issues into more manageable ones. To achieve this, we devote considerable time to understanding what characteristics make these simpler problems solvable.

So, let’s delve into that process.

We’re going to analyze the solution to this equation. Why, you might wonder? (That’s a fair question.)

If we examine a basic equation like

and identify a concise list of key properties that enable this equation to be solvable, it could prove quite beneficial. Specifically, these fundamental properties could help us define an abstract “algebraic structure” for broader study. In the end, this exploration will assist us in addressing more complex equations.

Now, let’s imagine for a moment that the solution “x = 1” did not immediately come to mind. What would your next step be?

We seek a number that, when added to 2, equals 3. Thus, subtracting 2 from 3 could be a logical maneuver… And indeed, that leads us to x = 1.

In high school algebra, this method is termed “subtracting the same value from both sides” of an equation. (You could also “add the same value to both sides.”)

Remember: We’re dissecting this process, so we’ll proceed at a deliberately slow pace.

If x + 2 equals 3, it follows that subtracting 2 from both sides should yield two equal values. Yet, we won’t alter the equation further for now. Our equation thus transforms into:

Numbers? What Numbers?

Before we advance, we need to clarify some assumptions. I jumped straight into an equation, and due to its simplicity, we made a few assumptions, you and I.

Firstly, the numbers we’ve chosen appear to be straightforward whole numbers—no troublesome fractions or Greek letters involved.

Mathematicians refer to these whole numbers as integers. This includes 0, 1, 2, 3, 4, etc. However, it also encompasses negative whole numbers like -1, -2, -3, -4, etc.

So, we are concentrating on equations that involve integers. Excellent.

What is Subtraction, Really?

Let’s examine another aspect of our original equation.

The “operation” at play in this equation is addition. Therefore, we’re focusing on integers and the process of adding them.

However, we’ve also begun discussing subtraction. Let’s analyze that for a moment.

What is the connection between subtraction and addition? They are akin to siblings, almost twins. I might even argue that subtraction is just a form of addition since I can convert any subtraction problem involving integers into an addition problem.

For instance, 3 - 2 is equivalent to 3 + (-2), correct? In other words, subtracting 2 from 3 is the same as adding 3 to -2.

Consequently, if we aim to distill the solution of this equation to its core components, we don’t actually need both the “subtraction” and “addition” operations. We can simply use addition, employing negative numbers as necessary.

Returning to our slow-paced equation-solving, we previously stopped here:

On the right side, we can perform the arithmetic and determine that it equals 1. On the left side, we will substitute subtraction with addition, resulting in:

This is where the magic starts. How do we know that the left side of this equation simplifies to just x?

You might want to jump ahead and say it’s because you recognize that 2 + (-2) = 0; and that is indeed correct. However, the three essential properties we need for solving equations are embedded within the leap you just made.

So, we will take it s-l-o-w…

Essential Property #1: Associativity

Mathematicians enjoy naming concepts. The rule that allows us to replace

with

is referred to as the _associativity_ property.

This is one of the three fundamental algebraic properties necessary for solving basic equations. Fortunately, addition of integers exhibits this associativity property. This means that when you are adding numbers, you can rearrange the parentheses in any manner, and the outcome will remain consistent, regardless of the “order of operation” used for the various additions. Therefore, it is entirely valid to transition from

to

Essential Property #2: Identity

The next essential property is subtly integrated into our process. The concept of “subtracting 2” (which we substituted with “adding -2”) to both sides of the equation was to allow us to isolate x on one side, ultimately leading to a solution.

Why were we confident we could execute this step? Well, we chose which number to add to both sides based on the knowledge that 0 (zero) behaves differently from all other integers.

Zero is the integer that essentially leaves others unaffected. When you add 0 to any value, it remains unchanged. Again, mathematicians enjoy naming concepts, and we refer to this as an _identity_.

Integers possess this special number 0, which is highly beneficial for solving addition-based equations.

Essential Property #3: Inverses

We have arrived at the third (and final!) essential property for solving basic equations. Beginning with the number 2, we identified another number (-2) such that their sum yields that special “identity” number 0. This allows us to transition from

to

And then, since 0 remains 0 (meaning, adding x to it returns x), we finally arrive at:

Groups

That was quite a bit of analysis over a seemingly simple equation, wouldn’t you agree?

Here’s the reward: we now have a concise list of properties necessary for solving basic equations that involve just one “operation” (in our case, addition).

We could have performed a similar breakdown if we examined an equation focusing solely on multiplication, leading us to the same three essential properties. (Although the identity number for multiplication is 1, not 0; and inverses manifest differently; and the sets of numbers involved would differ too — but the essential properties for solving equations remain the same!)

Now we transition into mathematical abstraction—

Imagine you have any set of objects along with an operation applicable to those objects, provided that the only other requirement is our three essential properties (the associativity rule, the existence of an identity, and the existence of inverses), then we refer to that set and operation as a group.

We can then investigate all the mathematical implications stemming from that fundamental definition. This field of study is called group theory and represents one of the foundational pillars of abstract algebra.

There are alternative methods to approach the study of groups, but I believe that deconstructing a simple equation captures the essence of abstract algebraic thinking.

Algebraists begin with basic concepts, distill them to their most fundamental essence, and then study them thoroughly.

If you’re still with me, you may also be an algebraist. Welcome! :)

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