Group theory

A group is a non-empty set with an operation that satisfies the following constraints: the operation is associative, has an identity element, and every element of the set has an inverse element.

Groups are the algebraic abstraction of symmetry, and symmetries are concrete realizations of groups.

Formal definitions

A group $G$ is a set with a binary operation $\cdot$ such that $\mathrm{\forall }a,b\in G$:

1. Closure: $a\cdot b\in G$
2. Associativity: $(a\cdot b)\cdot c=a\cdot (b\cdot c)$
3. Identity: $\exists e\in G$ with $e\cdot a=a\cdot e=a$
4. Inverse: $\forall a\in G,\exists a^{-1}\in G$ with $a\cdot a^{-1}=e$

Let $X$ be some object (a set, geometric figure, space, etc.). A symmetry is a bijection $f:X\to X$ that preserves structure (e.g., distances, angles, algebraic relations, etc.). The set of all such symmetries $Sym(X)$ forms a group under composition of functions.

Why symmetries always form a group

Take symmetries as transformations $f,g:X\to X$:

• Closure: composing two symmetries is still a symmetry
• Associativity: function composition is associative
• Identity: the identity map does nothing
• Inverse: every symmetry can be undone (bijection ⇒ invertible)

So:

Symmetries automatically satisfy the group axioms.

The converse also holds:

Every group is (isomorphic to) a group of symmetries.

This is known as Cayley’s Theorem. In other words, For any group $G$, you can represent its elements as permutations of $G$ itself:

So every abstract group can be realized concretely as symmetries of a set.