Abstract Branch of Math Covering Symmetry?

Question

If anyone could point me in the direction of a branch of mathematics that focuses on symmetry (or abstracting symmetry), I would greatly appreciate it. Not just in terms of functions and shapes, but at a higher-level than that.

Answer 1

Group theory, and more generally representation theory, deals with symmetry. The set of all automorphisms of a given object (a set, geometric space, algebraic structure) form what's known as a group. Concretely, a group is a set $G$ together with a multiplication map $\mu: G \times G \to G$ such that for all $a,b,c \in G$, we have $(ab)c=a(bc)$ ($\mu$ is associative). Furthermore we require that $\mu$ has an identity as well as inverses for all $x \in G$ (denoted $e$ and $x^{-1}$).

A group $G$ may act on an object $X$, and when it does we say that $X$ has $G$ as a symmetry group. An action is essentially a map $G \times X \to X$ that preserves any structure that $X$ may possess (alternatively, an action is a homomorphism (or map of groups) $G \to Aut(X)$). Perhaps the most important objects that groups act on are vector spaces, and structure preserving actions of groups on vector spaces is the object of study of the representation theory of groups.

One may also study objects related to groups (lie algebras, group algebras, quantum groups) and how they act on things, although interpreting these actions as symmetry is a little bit less straight forward.

If you'd like to read more on the subject, I'd recommend any introductory textbook on abstract algebra (and group theory in particular), and after that an introductory book on representation theory of finite groups (e.g. Serre or Fulton).