What is Group Theory?

If you're studying undergrad math, physics or chemistry, chances are you've heard of this thing called a "Group" that is studied in "Group Theory". What is it and why is it so important? I'll explain in a simple way.

The most familiar group is our number system; we have a bunch of numbers, and we have this thing called "$+$" which can take two numbers and churn out another number. A group is just a set of objects, and you have one "operation" (usually called "$\ast$" or "$+$", but those are just names) that tells you how to combine objects to produce other objects in that set, subject to a few rules. Why devise such a "strange chasing game" in a set of objects? That's because such systems are omnipresent in a stupendous array of phenomena:

• Most number systems (whole numbers, fractions, real numbers, complex numbers...)
  
• "Weird" number systems that can have strange rules like $1 + 1 = 0$ are used in binary computation, and are crucial in encrypting information for secrecy. Similarly, the "clock algebra" has $2359 + 0001 = 0000$.

Clock algebra: the numbers "reset" after advancing past a certain limit, like 24 hours. (Image by Martin Pettitt)

• Transformation groups - transformations of space, like a rotation $A$ or a reflection $B$ etc, can be combined into new transformations by defining $B \ast A$ to mean "do $A$ then do $B$". This is absolutely fundamental in physics, from the physical laws of "our space" to investigating the laws of "new spaces", like Quantum Mechanics and Relativity were; and in chemistry, from studying crystals to calculating molecular orbitals. Check "Noether's Theorem" for more.

Ice is made of water molecules arranged in a regular lattice. Its symmetry can be characterized by the transformations of space that leave the lattice unchanged. (Image generated by a program I wrote)

• The Rubik's Cube Group - every possible configuration is a combination of the basic moves (read the moves from right to left). In the figure below,

1. $\mathrm{X1} = \mathsf{NoMove}$
2. $\mathrm{X2} = \mathsf{BackTwist}$
3. $\mathrm{X3} = \mathsf{TopTwist} \ast\mathsf{BackTwist}$ (read from right to left, so Back Twist then Top Twist)
4. $\mathrm{X4} = \mathsf{LeftTwist} \ast\mathsf{TopTwist} \ast\mathsf{BackTwist}$

I know that each twist can go in two opposing directions, but you get the point.

Each configuration of the Rubik's Cube can be described by the sequence of moves used to reach it. (Image by Tom Davis)

• Pretty much every "algebraic system" is built on special kinds of groups - e.g. groups of vectors, groups of matrices, groups of polynomials...

\begin{align}
\begin{bmatrix}
1\\2\\3
\end{bmatrix}+\begin{bmatrix}
4\\5\\6
\end{bmatrix}&&
\begin{bmatrix}
1&0&0\\0&1&0\\0&0&1
\end{bmatrix}+
\begin{bmatrix}
0&1&0\\0&0&1\\1&0&0
\end{bmatrix}&&
\mathsf{Quadratic} + \mathsf{Cubic}

\end{align}

• Many, many, many more...

This exemplifies one power of Mathematics, where a myriad of disparate phenomena and concepts can have a single simple abstract underpinning. Group Theory is the study of that underpinning, thus its results and implications reach far and wide.

Technical Note

When I said "subject to a few rules", I was referring to the following rules that hold for any group with object set $G$ and operation $\ast$:

1. Associativity. Any objects $a$, $b$ and $c$ in $G$ must satisfy $(a \ast b) \ast c = a \ast (b \ast c)$. You can verify that this works in the above examples—another "strange" abstract underpinning!
2. Identity. $G$ must contain a special object $e$ called the "identity" so that $e \ast a = a = a \ast e$ for every object $a$ in $G$. When adding numbers, $e = 0$ so $0 + a = a = a + 0$. For the Rubik's Cube, $e = \mathsf{NoMove}$ doesn't change the cube, and performing $\mathsf{TopTwist}$ before or after a $\mathsf{NoMove}$ is the same as doing just the $\mathsf{TopTwist}$.
3. Inverse. Each object $a$ in $G$ must correspond to an "inverse" $b$ so that $a \ast b = e = b \ast a$. When adding numbers, the inverse of $a$ is $-a$ so that $a + (-a) = 0 = (-a) + a$. For the Rubik's Cube, the inverse of each twist is the same twist but in the opposite direction; $\mathsf{TopTwist} \ast \mathsf{TopOppositeTwist} = \mathsf{NoMove}$ $=$ $\mathsf{TopOppositeTwist} \ast \mathsf{TopTwist}$.