Introduction to Group Theory

Think about the world around you for a second. You know, from like the giant swirls of a galaxy way out there down to the tiny, perfect structure of a crystal , or even just the patterns you see in buildings. Yeah, there's this amazing regularity, isn't there? Exactly. A kind of order, a structure. Maybe you call it symmetry. It reminds me of that quote from D'Arcy Thompson, the biologist: "The beauty of a snow crystal depends on its mathematical regularity and symmetry."  (Read about D'Arcy Wentworth Thompson's theory on how physical and mathematical principles shape the growth and structure of living organisms. Instead of focusing solely on Darwinian natural selection, Thompson emphasized the role of physical laws and geometry in biological form and development.)
See, this is going to be in informal way, but that doesn't mean you do not read it carefully. Please read it very carefully it will help you in building a good foundation on Group Theory.

First Example - A Tetrahedron

Let's start with a concrete example: A Regular Tetrahedron (pyramid shape with four identical triangle faces – equilateral triangles, all edges the same length .
Now, before talking about it's symmetry in detail, let me briefly define symmetry: Imagine you have an object, like a perfectly square tile or a beautiful snowflake. A symmetry is a transformation (like a flip, a turn, or even just leaving it alone) that you can do to the object, and after you're done, it looks exactly the same as when you started. It fits perfectly back into its original outline or space. It's like if you pick it up, spin it around somehow, put it back down, I wouldn't know you'd done anything unless I'd put some mark on one corner. 

Pick a tetrahedron by one of its vertices (the corners), and imagine an axis going straight through one corner (vertex) and piercing the middle of the opposite triangular face. You now spin the shape around that axis.
Here’s what you will get:

• A rotation of 120 degrees (that’s 2π/3 radians) keeps the shape looking identical.

• Do another spin, same amount—240 degrees or 4π/3 radians​—still looks the same.

• Do it a third time (360 degrees) and, well, you’re back where you started.

And because there are 4 vertices, that gives us 4 axes, each allowing two non-trivial rotations (120° and 240°), plus the identity (doing nothing at all). So far, that makes the total: 4 × 2 = 8 distinct rotations

(We’re not double-counting the 360° spins because those are basically just the “do-nothing” move.)

Now take two edges that don’t touch each other (non-adjacent)—there are exactly three such pairs in a tetrahedron. Connect their midpoints with another imaginary axis. These axes run through the middle of the tetrahedron, and they let you spin it 180 degrees (aka π radians). And guess what? That spin also brings you back at something that looks exactly like the original tetrahedron. So you get:
3 more rotations, one for each edge-edge axis.

Now we count all the rotation, counting the do nothing operation once, we get total: 8+3+1 = 12 rotational symmetry.
Now here comes an interesting part, Suppose you take one of these 12 rotations and then, apply another one. what would be the result?
It’s always going to be from one of those 12 rotations we have already discussed. You cannot somehow create a 13th way! And this is what is called closure. The set is locked in—it’s a closed system under composition (formal word for doing an operation after some other operation).

https://www.geogebra.org/m/nph3q5rv
(In this not all rotation symmetry is shown, but it is sufficient for understanding purpose - Play with this for a while till you get a hang of it)

Symmetry of Equilateral Triangle

Now we will define what is a group in a more formal way. But what is the need, actually once we define a group in generality we will be in a position to identify groups easily. There may be other collection of objects that have the structure of the group and they maybe not even geometric shapes. To classify them formally we need a general definition.

Okay, so a group is fundamentally two things:

1. A set of elements (G). These could be rotations, numbers, matrices, whatever.

2. An operation ('◦') to combine any two elements  from G to get another element, which must also be in G (closure!). - Binary Operation

This set and binary operation combo has to follow just three specific rules (axioms):

1. Associativity: If you're combining three elements (a,b,c) in order, it doesn't matter how you group them: (a◦b)◦c=a◦(b◦c). The parentheses can move, but the order can't.

2. Identity Element: There must be one special element (e) in G that doesn't change anything: a◦e=e◦a=a. For our tetrahedron, it was the "do-nothing" rotation.

3. Inverse Element: For every element x in G, there must be another element (y, often written x−1) in G that "undoes" it: x◦y=y◦x=e. For a rotation, it's rotating back.

So, you see, the very nature of symmetries—how they combine, how they can be undone, the existence of a "do nothing" option—lines up perfectly with the formal axioms of a group! In a way, the abstract definition of a group is inspired by these concrete properties of symmetries. 

Symmetry/Permutation Group

Okay so all the examples that I have discussed so far had geometric structure and some of their symmetries were easily visualised. Now I would like to discuss a very important symmetry group, which is very general in nature and also there is famous theorem that shows its generality.
Let’s talk about one of the most general and powerful types of symmetry: permutations—which just means “rearrangements.” 

What’s a Permutation?

Think of a set of distinct items. Like 3 colored balls:

• 🔴 Red

• 🟢 Green

• 🔵 Blue
  
  

Each ball sits in its own spot. A permutation is just a way to shuffle these items around. You’re not adding or removing anything—you’re just reordering them. (more formally a permutation is a bijective map from a set to itself)

Let’s imagine this simple setup: A collection of 6 distinct points, sitting at fixed positions on a plane.

We’re not talking about a shape like a triangle or hexagon with straight sides—just 6 separate, labeled points, like pins on a board. Their positions are fixed, but what we’ll play with is which label (or identity) is where.

Now imagine we relabel or swap the points—shuffle them around without changing where the actual positions are. What you’re doing is:

• Reassigning labels to the fixed spots

• But you’re not changing the overall structure—all 6 spots are still filled, just their labels have changed.
  
  

Since all the points are distinct, every possible way of rearranging their labels is allowed, as long as each point ends up with a new unique label (or the same, if nothing changes). And each way to rearrange them is not keeping the structure, therefore all these possible rearrangements form a symmetry of this configuration.  (Note: We’re thinking of the entire set as the object, not how it's drawn or how the points are connected, and this is the reason it is  very fundamental in nature and once we get a good grasp of it you’re learning about the core idea of symmetry itself. And thanks to Cayley’s Theorem, you’re also secretly learning about every possible group, even the ones you haven't come across yet.

Keeping all this aside first let us find how many elements are there in this group and what is the group operation. To answer the first question we need to answer, in how many ways can we rearrange 6 distinct labels?

That’s just the number of permutations of 6 elements = 6! = 6 × 5 × 4 × 3 × 2 × 1 = 720

So there are 720 possible symmetries of this labeled set of 6 points.
This collection of 720 permutations forms a group, under the operation of composition ( do one permutation, then another and the result is still a valid permutation). It is easy to verify that the set of these permutations along with the composition operation satisfies all the group properties  and this group is called the Symmetry group Sn

There's a super important idea in group theory called Cayley's Theorem. It basically says that every single group, no matter how abstract or weird it seems, is structurally the same as (isomorphic to) a subgroup (a part) of some Symmetric group Sn​. 

Some Other Examples of Groups

Number Systems as Groups 

Let's go back to shapes, but this time we will also include reflections. The dihedral group Dn is the group of all symmetries (rotations and reflections) of a regular n-sided polygon. Consider the hexagon (D6). It has:

• 6 Rotations: 0°, 60°, 120°, 180°, 240°, 300° (R0, R60, … etc.)

• 6 Reflections: 3 through opposite corners, 3 through midpoints of opposite sides (let's call them S1, S2 , … etc.)

That's a total of 12 symmetries. (Hey, same number as the tetrahedron rotations!!).

Let R be the smallest rotation (60°) and S be one specific reflection. We know R6

=e (rotate 6 times = do nothing) and S2

=e (flip twice = do nothing). But what happens when you mix them?

It turns out there's a crucial relationship: S◦R=R−1

◦S. This means: "Flip then rotate" is the same as "Rotate backwards, then flip".

Crucially, this means S◦R

=R◦S (unless R is 180° or 0°). If you flip the hexagon and then rotate it, it lands in a different position than if you rotate it first and then flip it. The order matters!

Let R be the smallest rotation (60°) and S be one specific reflection, we know that

But what happens when you mix them?
It turns out there's an important relationship between R and S:

The General Linear Group GL(n,R)