Group Theory

Take a Rubik’s Cube. You can turn one side, then another, and you can undo every turn. No matter how often you turn it: you always end up with a cube again, never with something else. Group theory is the part of mathematics that studies exactly such systems of operations. It doesn’t ask what individual numbers look like, but how operations can be combined. A group is nothing other than a set of such operations together with a rule for how to carry out two of them in sequence.

Why symmetry can be calculated

The great trick of group theory is abstraction. The rotations of a cube, the shuffling of a deck of cards, and the addition of clock times look completely different. Mathematically, however, they obey the same four basic rules. Anyone who proves something about these rules once has proven it for all these cases at the same time.

That is why group theory has become a kind of universal language for symmetry. In physics, it describes which quantities remain unchanged during a process. The Standard Model of particle physics, i.e. the best known description of the smallest building blocks of matter, is at its core formulated in the language of group theory. The classification of crystals in chemistry is also based on it.

For technology and business, a different point is more decisive. Almost every modern encryption computes in groups. When your browser secures a connection to a bank, it performs calculations in the background whose security relies on group-theoretical properties. Without this field, there would be no secure online banking.

The four rules that make up a group

A group consists of a set of elements and an operation that turns two elements into a third. First, the result must always lie within the set again. Second, the placement of brackets must not matter. Third, there must be a neutral element that changes nothing. Fourth, every element needs a counterpart that cancels out its effect.

An everyday example is the clock. Add hours on a twelve-hour dial: 9 plus 5 equals 2. The result always stays between 1 and 12, i.e. within the set. The 12 acts as neutral, because 12 hours later the clock shows the same thing. And for every shift there is a counter-shift. This structure is called modular arithmetic and is one of the most important groups of all.

A common misconception: that in a group, order must not matter. That only holds for so-called commutative groups. With the Rubik’s Cube, it’s different. Turning the top first and then the right side gives a different result than the other way around. It is precisely this non-commutativity that makes many groups interesting and mathematically difficult.

From bank connections to Rubik’s Cube records

The most direct encounter with group theory is in cryptography, the science of encryption. Methods like elliptic curve cryptography use a group in which computing forward is easy and computing backward is practically impossible. These methods are found in chip cards, messengers, and cryptocurrencies. When news talks about post-quantum cryptography, it’s about which groups a quantum computer could crack.

A second field is error correction. Data on a hard drive or in a radio signal is encoded so that corrupted bits can be detected and repaired. The underlying codes are constructed using group theory. Quantum computers also need such codes, because their computational building blocks are extremely prone to disturbance.

In computer science itself, group theory helps with counting. It answers questions like: how many truly distinct positions does a Rubik’s Cube have? The answer, around 43 quintillion, comes from a group-theoretical calculation. The concept is also used in machine learning: models that have a problem’s symmetries built in need less training data. Such approaches fall under the term geometric deep learning.