# Definition:Group Action/Right Group Action

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## Definition

Let $X$ be a set.

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

A **right group action** is a mapping $\phi: X \times G \to X$ such that:

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

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

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

\((\text {RGA} 2)\) | $:$ | \(\ds \forall x \in X:\) | \(\ds x * e = x \) |

## 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.