Motivation and definition

An action of a group $G$ on a non-empty set $X$ is:

A map $\cdot: G\times X \to X$, $(g,x) \mapsto g\cdot x$, such that

$1 \cdot x = x. \forall x \in X$

$(gh) \cdot x = g\cdot ( h \cdot x)$

Example: Left translation action

With a group action, every element in a group defines a bijection on the set acted upon, to itself, defined just by acting with that group element. This can be used to proove Cayley's theorem

Transitive group action

Group orbit

Stabilizer subgroup

Conjugated, Conjugation action

Group action on power set

https://en.wikipedia.org/wiki/Group_action