Action

Let $M(X)$ denote the Group of all invertible Maps $X\to X$ and let $G$ be any Group. A Homomorphism $\theta:G\to M(X)$ is called an action of $G$ on $X$. Therefore, $\theta$ satisfies

1. For each $g\in G$, $\theta(g)$ is a Map $X\to X:x\mapsto \theta(g)x$,

2. $\theta(gh)x=\theta(g)(\theta(h)x)$,

3. $\theta(e)x=x$, where $e$ is the group identity in $G$,

4. $\theta(g^{-1})x=\theta(g)^{-1}x$.

See also Cascade, Flow, Semiflow