Understanding Operations in Group Theory

What’s dark chocolate?

Dark chocolate is a form of chocolate made from…

Can you understand what dark chocolate is if you don’t know anything about chocolate? That’s the same thing for operations in group theory.

A group is a set equipped with a binary operation that satisfies four axioms. Like the definition of dark chocolate, the notion of operation is coupled with the definition of a group. Learn about operations and sharpen your knowledge of groups for free!

An operation must satisfy three essential properties: being well-defined, closure, and totality (being defined for all elements). We could simply memorize those properties as a gift from those brilliant mathematicians fortunate enough to define mathematics, but that won’t build our understanding. Why does each property exist, and what happens when we remove them? Those questions are better candidates for the purpose of comprehension.

An operation must be well-defined

If an operation is a machine that takes an input and produces an output, a well-defined operation gives out only one output for any given input.

Consider the simple equation:

$$x + 2 = 3$$

To solve for $x$, we subtract 2 from both sides, isolating $x$ on the left side as a result:

$$ x = 3 - 2 $$

Since the operation $3 - 2$ gives a unique answer, we can confidently conclude that $x = 1$.

Now, imagine if the result of $3 - 2$ was not unique. What if $3 - 2$ sometimes equals $100$ or perhaps $4234$? When would it be $100$? Equality assumes uniqueness. Without it, mathematics would become chaotic. We couldn’t write equations or prove theorems, as the results of our calculations would be constantly changing.

An operation must be closed

Closure ensures that when we perform an operation on elements within our set, the result remains within the same set.

Consider an analog clock. Our operation will be addition of hours on a clock. We know that the clock can’t go higher than 12 hours. The time wraps around at 12.

If you start at 8 and add 2 hours, you end up at 10. If you add 6 more hours, you reach 4 (since $(10 + 6) \bmod 12 = 16 \bmod 12 = 4$). No matter which numbers you combine using this addition, you will always stay within the set $\{1, 2, 3, ..., 12\}$, which demonstrates closure visually.

Were addition not closed, what would $8 + 6 + 9$ equal? Instead of getting 11 (since $(8 + 6 + 9) \bmod 12 = 23 \bmod 12 = 11$), we might get some value outside our clock’s range, like 23. In the context of the clock, does 23 hours have any meaning? When the operation is not closed, you lose consistency. You aren’t guaranteed to chain operations and get a meaningful result within your system. Why should we value consistency? Because it guards us against extraneous element within our system.

An operation must be total (defined everywhere)

Finally, an operation must be defined for all pairs of elements in the set - this property is called totality. If we have a set $S$ and an operation $\oplus$, then for any two elements $a$ and $b$ in $S$, the operation $a \oplus b$ must be defined and produce a result.

Consider the division operation on real numbers. Division is not total because dividing by zero is undefined. If we tried to use division as our operation on the set of all real numbers, we would have gaps - certain combinations of elements (like $5 \div 0$) would have no result.

This violates our need for consistency that we saw with closure. If we allow undefined operations within our set, it weakens our ability to describe that set consistently, which means having to deal with exceptions to statements. We can no longer say “any two elements can be combined using our operation” - we’d have to add caveats and exceptions.

Totality ensures that our operation is a true function that works for every possible input from our set, making our mathematical system complete and predictable.