Definition:Group Action

Also defined as
A 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 $\phi_g \left({x}\right)$ 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