Groups

We're all familiar with the idea of adding/multiplying numbers together.$3 + 2 = 5$, $3 \times 2 = 6$ etc.

Group theory concerns the study these operations in a more abstract setting. Instead of just numbers, we can have any kind of object and any kind of operation. The only requirement is that the operation satisfies a few properties. We'll cover these properties later, but first let's look at an example.

The Triangle Group

The most classic group example is the rotations and reflections of a triangle.

If we label the vertices of our triangle $a$, $b$ and $c$, then any specific rotation or reflection of the triangle can be represented by a permutation of these vertices. Here's how that works:

As you can see as we rotate the triangle we can represent its state by the permutation of the vertices. Rotating the triangle changes the permutation.

We can also perform another operation on the triangle: flipping it along the vertical axis. We can see this operator as well with the permutations:

Now that we've got a visual intuition let's formalise this in mathematical terms.

We'll create a group out of this by considering how these operations combine. We'll call our group $G$ and the elements of the group will be all the operations we can perform on the triangle. There are 3 different operations we can perform: a 120 degree rotation, a 240 degree rotation and a flip. We'll represent these as $r_1$, $r_2$, $f$ respectively. In fact, we'll add a 4th operation here as well: doing nothing. We'll call this the identity operation and represent it as $e$.

The last thing to cover is how these operations combine. We'll use the symbol $\circ$ to combine our operations. And now we're ready to do some math!

It's quite clear that $r_1 \circ e = r_1$, $f \circ e = f$ since doing nothing (the identity operation $e$) after any operation will be the same as just doing the operation. We can see some more interesting properties as well. For example $r_1 \circ r_2 = e$. If we do a 120 degree rotation followed by a 240 degree rotation we end up back where we started. This is the same as doing nothing.

We can create something very similar to a multiplication table showing how the $\circ$ operation works between each element:

Ok, we've got a few interesting values in the table, but what about those question mark values? What do we get when we flip the triangle and then rotate it? If we start with a triangle permutation $ABC$, flipping then rotating gives us:$BAC$. We can't get to this state with a single flip or rotation so we've found a new operation that we need to add! We'll call this operation $fr_1$. We'll also need to add$fr_2$ to represent the state $CBA$. Now our table can be completed:

I encourage you to get a pen and paper and check that some of these values are correct.

This is a group! We have a set of elements and a binary operation that combines them. Formally we define a group as $(G,\circ)$ where $G$ is our set of elements and $\circ$ is the binary operator the combines the elements. So in our case: $G = \{e, r_1, r_2, f, fr_1, fr_2\}$ and $\circ$ is the operation defined in the table above.

The essence of a group really lies in the multiplication table we created. Given any number of elements simply draw a table and fill in rows and columns and you've got a group!

Or have you...?

There are actually a few rules that must be followed in order for it to be a group. We can't just enter any values we want into the table. For example a group of 3 elements table where the operation $\circ$ sends every value to $e$ (the identity) is not a valid group.

Let's go through each of the rules now and relate them to our triangle group example.

Closure

The closure property states that when combining any 2 elements of the group under the operation, the result is also a member of the group.

This is a not a tricky one to prove. Simply check the multiplication table above and you'll see that the result of any $\circ$ operation gives another group element.

Associativity

Just like normal numerical multiplication, when combining elements of the group, we must be able to pick any ordering of brackets we want.

Proving this for our triangle group will be a little more involved but I encourage you to try a few examples and check that it holds e.g. $(r_1 \circ fr_1) \circ f = r_1 \circ (fr_1 \circ f)$

Identity

The math notation is getting a bit more complicated now, but this one's simple. It merely states than a group must have an identity element. i.e. the "do nothing" element.

This one's easy to prove as well since we can quite clearly see from the left and top row of the table that $e$ is indeed an identity.

Inverse

For every element of the group there must exist some other element that when combined with that element produces the identity. Think about numerical multiplication where every number has an multiplicative inverse that when multiplied together gives $1$, e.g. $5 \times \frac{1}{5} = 1$

We can see this is true from the fact that each row/column of the table has exactly one $e$ in it.

Well-Defined

This one seems obvious, but it states that if two equal elements are combined in the same order under the operation they give the same result.

This is trivially true for the triangle group since we can see each entry in the table has only one entry.

And these are the 5 rules! I encourage you to have a go at constructing some other multiplication tables of groups and see if they fit the rules.

One interesting rule you might notice is missing is commutativity. Commutativity states that $a \circ b = b \circ a$. This is not actually a requirement of a group. The triangle group we've been discussing is not commutative! The table above shows that $fr_1 \circ r_1 \neq r_1 \circ fr_1$. If a group does have this property we call it an Abelian group.

Conclusion

One final question you might ask here is: why these rules? Why do we need these specific properties to define a group? What would happen if we removed some? The way to answer this question is to simply state that these rules are chosen because they are interesting. There is nothing that makes these rules "right" or "correct", but they are the rules that give us the most interesting and useful properties for the math we need. Changing the rules would give us a different structure that might be interesting in its own right, but it wouldn't be a group.

This is the essence of group theory. We're not interested in the specific operations we're performing, but the properties that these operations satisfy. This allows us to study the abstract properties of these operations and apply them to many different areas of mathematics.

Exercises