Groups are a common object in abstract mathematics, but they also have various applications outside of math as we will see later. In this post I will give you the details of what a group is and because theoretical definitions can be hard to grasp, I also included a simple example.
Definition of a group
A group in mathematics consists of two parts: a set of elements called \textbf{G} from here on and a binary operation, which is an operation that takes two inputs. One example for a binary operation would be basic addition as most of us know it: a + b. This operation has two inputs, one on each side of the plus sign. You could also write it as a function, similar to f(x) which we see so often, this could look like this: +(a, b) or add(a, b).
The most common notation for this operator when talking about groups is similar to the notation of the multiplication of vectors for example: you simply write both elements next to each other, like ab is the application of the operation to the elements a and b that are both in the set G.
The following three properties must hold for G with its operation to be a group:
- G must be closed under its operation – meaning for every a and b from G, the element ab has to be in G as well.
- the operation is associative, meaning (ab)c = a(bc) for all elements a, b, c in G.
- the operation guarantees an identity element, meaning in G there exists an element e, which will be called the “identity element”, with ae = ea = a for all elements a \in G.
- the operation guarantees inverse elements, meaning for all a in G there exists an element x in G such that ax = xa = e. x is then called the inverse of a.
Note that we do not demand that ab is the same as ba. An operation that fulfills this property is called commutative and a group with such an operation is often called an abelian group or commutative group.
Example of the group \mathbb{Z}_n
An example for a group is \mathbb{Z}_n which consists of the set \{0,1, ..., n-1\} and the add operation modulo n. Let’s look at this with the example \mathbb{Z}_5 with the numbers \{0,1,2,3,4\} and the operation a +_5 b = (a+b) \text{ mod } 5 for all elements a, b \in \mathbb{Z}_5.
Example calculations: 3+_5 2 = 0 and 4 +_5 4 = 3.
This group is closed, because if we add modulo 5, we can not get out any number bigger than 4 by definition and by only adding numbers bigger or equal to 0 we can not get any negative numbers either.
This addition is associative simply by the rules of modular arithmetic – meaning the way modulo was defined. It is also commutative for the same reasons, the same way “normal” addition is commutative: a+b = b+a for all real numbers. This makes \mathbb{Z}_5 an abelian group.
The identity element in this group is 0 because a+ 0 = 0+a = a, which is true with and without modulo.
We can also find out the inverse elements by simply trying all pairs (or by using what we already know from basic addition). In the examples above we already saw that 3 is the inverse of 2, because if you add them, you get the identity element 0. Because the order does not matter, 2 is also the inverse of 3.
Furthermore, 4 +_5 1 = 0 as well and 0 +_5 0 = 0.
| Element of $G$ | Inverse of that element |
|---|---|
| 0 | 0 |
| 1 | 4 |
| 2 | 3 |
| 3 | 2 |
| 4 | 1 |
Applications for Group Theory
Group theory is the basis for many other mathematical areas such as algebraic topology and geometry which in turn have applications as well. The group is just a very basic mathematical object that is learned in the first semester of studying math usually, so it is hard to describe an immediate application of group theory.
However, they seem to be a few fields that can make immediate use of the studying of groups. As we saw above, symmetries are a great way to visualize groups and that is also what many applied fields deal with that draw on the theory of groups.
According to this https://en.wikipedia.org/wiki/Group_theory#Applications_of_group_theory, group theory is used in physics, chemistry and cryptography for example.
Physicists – and other scientists – probably disagree, but I have long suspected that physics is mostly applied mathematics. They seem to find applications for the most abstract stuff that mathematicians come up with in cryptic ways I will likely never understand. Group theory in physics is useful because symmetries of physical systems correspond to conservation laws.
In chemistry they use group theory to classify the symmetries of molecules and in cryptography very large groups of prime order are used in public-key cryptography.
Let me know in the comments, if you have any more questions about groups or if you can maybe tell me a bit more about applications of groups from your field of interests. I studied math more in its pure form, so I’m not as knowledgeable in its applications.