# Does group theory deal with asymmetry?

I have a question from which you should notice that I do not have much of a clue abot group theory. At least not yet.

The question is about that many introductory articles about group theory seem to refer to the use of group theory with rotations of bodies and their related symmetry.

What I am curious about is whether it is possible to use group theory on bodies that are asymmetrical or where the symmetry is broken (this latter could be always the same as asymmetrical, I am not sure yet)?

Thanks in advance

## Answers and Replies

When dealing with motions and symmetries of shapes, group theory is useful in that it allows you to do calculations related to those shapes and their symmetries. (For example, how many ways are there to colour a cube with n different colours? This is answered by (not) Burnside's lemma.) When you have an asymmetrical situation, group theory doesn't really have any application.

There's more application of group theory than symmetries of objects, though. Galois theory and algebraic topology come to mind.

Group theory is the study of any set with a binary operation (*) that is

associative: (a*b)*c = a*(b*c)
there is an identity e such that a*e = e*a =a
and every a has an inverse such that a*a^-1 =a^-1*a = e

Technically, you could very very deep into group theory without doing anything with symmetries. It just turns out that the discrete symmetry groups like rotations and reflections of a polygon and continuous symmetry groups like the rotations of a circle are very useful and come up often. Also, if you've taken linear algebra you know that a rotation can expressed as a matrix. So it comes as no surprise that these more abstract groups can be represented by matrices and the binary operation is just regular multiplication of matrices.