Group Action

A group action is a formal way to represent the elements of a group $G$ as symmetries or transformations of a set $X$. Formally, a group action is a function $f:G\times X\to X$ satisfying two axioms:

1. Identity: The identity element $e$ of $G$ acts as the identity on $X$: $e\cdot x=x$ for all $x\in X$.
2. Compatibility: For all $g,h\in G$ and all $x\in X$, $g\cdot (h\cdot x)=(gh)\cdot x$.

Examples

• Group of Permutations $\Sigma (S)$ naturally acts on S.
• A group can act over itself, simply by multiplying.
• There is the trivial/identity action that ignores the group member and returns the input.
• There is another action that could apply to a group over itself: $$ \begin{aligned} \mu :~ &G {\texttimes} G\to G \ & (x, y) \to x^- \star y \star x \end{aligned}

## See also - [[Orbit-Stabalizer Theorem]] - [[Group Theory]] - [[Group]] - [[Faithful Action]]