group action

Let $G$ be a group and let $X$ be a set. A left group action is a function $\cdot: G \times X \longrightarrow X$ such that:

1. $1_G \cdot x = x$ for all $x \in X$
2. $(g_1 g_2)\cdot x = g_1 \cdot (g_2 \cdot x)$ for all $g_1, g_2 \in G$ and $x \in X$

A right group action is a function $\cdot: X \times G \longrightarrow X$ such that:

1. $x \cdot 1_G = x$ for all $x \in X$
2. $x \cdot (g_1 g_2) = (x \cdot g_1) \cdot g_2$ for all $g_1, g_2 \in G$ and $x \in X$

There is a correspondence between left actions and right actions, given by associating the right action $x \cdot g$ with the left action $g \cdot x := x \cdot g^{-1}$ . In many (but not all) contexts, it is useful to identify right actions with their corresponding left actions, and speak only of left actions.

Special types of group actions

A left action is said to be effective, or faithful, if the function $x \mapsto g \cdot x$ is the identity function on $X$ only when $g = 1_G$ .

A left action is said to be transitive if, for every $x_1,x_2 \in X$ , there exists a group element $g \in G$ such that $g \cdot x_1 = x_2$ .

A left action is free if, for every $x \in X$ , the only element of $G$ that stabilizes $x$ is the identity; that is, $g \cdot x = x$ implies $g = 1_G$ .

Faithful, transitive, and free right actions are defined similarly.