Definition:Group Action

Definition

Let $\displaystyle X$ be a set.

Let $G$ be a group whose identity is $e$.

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

$\forall \left({g, x}\right) \in G \times X: \phi \left({\left({g, x}\right)}\right) \in X = g * x \in X$

in such a way that:

• GA-1: $\forall g, h \in G, x \in X: g * \left({h * x}\right) = \left({g h}\right) * x$;
• GA-2: $\forall x \in X: e * x = x$.

We say that the group $G$ acts on the set $X$.

Notes

Some sources call 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.

Sources

• J.A. Green: Sets and Groups (1965)... (previous)... (next): $\S 5.5$
• Allan Clark: Elements of Abstract Algebra (1971)... (previous)... (next): $\S 53$
• John F. Humphreys: A Course in Group Theory (1996): $\S 10$: Definition $10.1$