Group Action

Let $G$ be a group and let $X$ be a set . A (left) group action of $G$ on $X$ is a function $\alpha:G\times X\to X$ (usually written $\alpha(g,x)=g\cdot x$) such that:

1. $e\cdot x = x$ for all $x\in X$, where $e$ is the identity of $G$;
2. $(gh)\cdot x = g\cdot(h\cdot x)$ for all $g,h\in G$ and all $x\in X$.

Equivalently, an action is the same data as a group homomorphism from $G$ into the group of bijective self-maps of $X$ (permutations), i.e. a permutation representation . Actions organize many constructions (e.g. actions on cosets) and lead to the notions of orbits and stabilizers

Examples:

• (Left translation) $G$ acts on itself by $g\cdot x := gx$.
• (Action on cosets) If $H\le G$, then $G$ acts on the set of left cosets $G/H$ by $g\cdot (xH):=(gx)H$.
• (Conjugation) $G$ acts on itself by $g\cdot x := gxg^{-1}$.