Beautiful Equations Blog - An Introduction to Group Theory. A rubik's cube sits on an old book in front of a large bookcase.

Anyone who has studied maths at university will have realised quickly that this is not the same subject you learn at school. There are hints of more recognisable subjects, such as statistics and mechanic – however, from the very beginning of the course, you are also introduced to abstract algebra and suddenly fields, rings and groups take on very different meanings. Whether you’re about to start a maths degree and want to give yourself a head start or are just interested in learning something new here’s a short introduction to one of these areas of abstract studies, Group Theory, that we hope will spark your interest!

So to start, what is a group? Slightly unhelpfully a group can be whatever you want it to be... as long as it follows certain rules called axioms.

A group is a set, say G, of elements, combined with some binary operation such that

1) For all elements of the group, e.g a and b, the element produced as a result of applying the binary operation to a and b, ab, is also in the group.

2) For any three elements in the group, say a, b, c, then the element (ab)c must be equal to a(bc) (where ab is the element formed by combining a and b with the binary operation). This property may seem familiar and rightly so: it is called associativity and is something that binary operations we use every day, such as addition and multiplication, also satisfy.

3) A group must have an identity element, some element e such that for any element of the group, say a, ae = a = ea.

4) Every element in the group must have some corresponding ‘inverse element’ that it can be combined under the binary operation to give the identity element.

This set of rules may seem extremely abstract but there are in fact several groups that you have been using since kindergarten. For example, if we take the set of whole numbers or ‘integers’, denoted ℤ, and the binary operation addition, then it’s possible to show that {ℤ,+} is a group. By definition, adding together any two integers will also give an integer, addition is associative as stated above, 0 is an integer and any number add zero will return that same number (so zero is the identity element) and finally for every integer, say a, -a is also an integer and a + -a = 0 and so all the axioms are satisfied.

Groups can be far more abstract than this, however, and are not limited to numbers as their elements. Take the set G = {“cat”, “dog”}. Defining a binary operation ‘*’ as Cat*Cat = Cat, Dog*Dog = Cat and Cat*Dog = Dog*Cat = Dog then {G , * } is a group! To see why again you just have to work through the axioms.

Some of the most important applications of group theory come from these stranger creations. The most common example is the “symmetric groups”. Studying the symmetries of some n-sided regular polygon and defining it as a group has led to applications across other sciences, for example in the structure of a hydrogen atom. If you are fortunate enough to come across group theory then don’t be scared! Abstract algebra can be intimidating at first but once you get your head round it, a symmetry group does become just as clear and natural as the set of real numbers!

- Lucy Chats Maths

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