Definition:Group Action/Right Group Action

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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.

Also see