Definition:Group Action

Definition
Let $X$ be a set.

Let $\struct {G, \circ}$ be a group whose identity is $e$.

A (left) group action is an operation $\phi: G \times X \to X$ such that:


 * $\forall \tuple {g, x} \in G \times X: g * x := \map \phi {\tuple {g, x} } \in X$

in such a way that the group action axioms are satisfied:

The group $G$ thus acts on the set $X$.

The group $G$ can be referred to as the group of transformations, or a transformation group.

Also defined as
A (left) group action is sometimes defined as what on is called a permutation representation.

As shown in Correspondence Between Group Actions and Permutation Representations, there is a one-to-one correspondence between the two.

Also known as
Some sources call $*$ a $G$-action and such an $X$ a this a $G$-set.

Some sources use $g \wedge x$ for $g * x$, while some use $g \cdot x$.

Some sources introduce the concept with the notation $\map {\phi_g} x$ for $g * x$, before progressing to the latter notation.

There is little consistency in the literature; $*$ appears to be popular. $\wedge$ is not generally preferred, because its other uses are somewhat specialized.

Also see

 * Equivalence of Definitions of Group Action