# Definition:Group Action/Left 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:

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

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

 $(\text {GA} 1)$ $:$ $\ds \forall g, h \in G, x \in X:$ $\ds g * \paren {h * x} = \paren {g \circ h} * x$ $(\text {GA} 2)$ $:$ $\ds \forall x \in X:$ $\ds e * x = x$

## Different Approaches

During the course of an exposition in group theory, it is usual to define a group action as a left group action, without introducing the concept of a right group action.

It is apparent during the conventional development of the subject that there is rarely any need to discriminate between the two approaches.

Hence, on $\mathsf{Pr} \infty \mathsf{fWiki}$, we do not in general consider the right group action, and instead present results from the point of view of left group actions alone.