Group action

A group action is a function that exploits symmetries in a space using group properties.

Definition

A group action of $G$ on some object $X$ is a homomorphism into the endomorphism group of the object.

$$G⟶\text{End}(X)$$

Equivalent (somewhat easier to use) definition

A (left) group action of a group $G$ on a set $X$ is a function $a:G\times X⟶X$ that satisfies the following conditions: $a(e,x)=x$ and $a(g_1{g}_2,x)=a(g_2,a(g_1,x))\forall x\in X,\text{and }{g}_1,g_2\in G$

The set $X$ together with the action $a$ of $G$ on $X$ is called a G-set.

We can define a right action… but literally everything I have used it for uses left actions… so I’ll leave it for now.

Notation

Generally we denote action by putting the elements next to each other or with $\cdot$. For example, $g_1(g_{2}x)=(g_1{g}_2)x$ or $g_1\cdot (g_2\cdot x)=g_1{g}_2\cdot x$

Morphism between $G$-sets

If $X$ and $Y$ are two $G$-sets, a morphism from $X$ to $Y$ is a function $f:X\to Y$ such that

$f(g\cdot x)=g\cdot f(x)\forall g\in G,x\in X.$

Automorphism

An automorphism is a special group of morphisms from the $G$-set to itself. These are denoted

${\text{Aut}}_{G}X$