Groups: The basics

Group theory is the mathematics of symmetry and structure. On this page, find out what a group is and how to think about them.

What is a group?

By a group, mathematicians mean a collection of actions which we can do one after the other, each of which can be undone.

For example, imagine holding a top upright and spinning it on its axis. No matter how much you’ve spun the top before, you can always keep spinning the top. Any amount of spinning you do can also be undone by spinning the top the same amount the other way. These two features together mean that the collection of all the ways you can spin the top forms a group. 

As another example, imagine walking about in a huge field. After walking in any direction for a bit, you can always continue your walk by taking any other walk in any other direction. You can also walk back from any destination you’ve gotten to. The collection of all your walks therefore also forms a group.

Read more about the definition of a group in a mathematical context here:

Maths in a minute: Groups — Look here for the mathematical definition of a group, along with plenty of mathematical examples!

Groups and symmetry

What do we mean when we say group theory is the study of symmetry? 

By a symmetry of an object, mathematicians mean some action we can do to the object which leaves all its features unchanged. For example, spinning a top about its axis is a symmetry of the top. 

Now take an object and consider the collection of all its symmetries. After each symmetry we can always do another symmetry. We can always also undo any symmetry we apply. The set of symmetries of an object always forms a group!

On the other hand, it turns out that any group can always be thought of as the symmetries of some object (called its Cayley graph).

So starting from an object, the set of its symmetries forms a group. And if we start from a group, we can always find it as the set of symmetries of some object. Hence the study of groups is exactly the same as the study of symmetries! This is what people mean when they say that group theory is the study of symmetry.

Groups as actions vs groups in themselves

Nowadays, it is very common to see groups not as collections of symmetries or actions, but as objects in their own right. In fact, most of the time when you open a textbook on groups, you’ll see a group defined as a set equipped with a binary operation (read more in Maths in a Minute: Groups to find out what this means).

We can therefore see groups in two different ways. Just as a word can be both a verb or a noun (to record music vs to play a music record), we can think of a group as either acting by symmetries on an object (a verb) or as an object in its own right (as a noun).

Translating from groups as acting on an object (groups as verbs) to groups as objects in their own right (groups as nouns) was the step into abstraction that first started group theory. We can also go the other way: starting from a group as an object in its own right (a noun), we can make the group act by symmetries on an existing object (a verb) using the notion of a group action.

Maths in a minute: Group actions — What exactly does it mean for a group to be a symmetry group?

What next?

Alongside concepts such as calculus and linear algebra, groups have become a core part of the language of modern mathematics and physics. Find out in the collection below how groups can help describe roots of polynomials, holes on a surface, and even the laws of physics.

Groups today: a whistle-stop tour — Find out how groups are used across mathematics and physics.