Definition:Group Action

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Definition

Let $X$ be a set.

Let $\left({G, \circ}\right)$ be a group whose identity is $e$.


A (left) group action is an operation $\phi: G \times X \to X$ such that:

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

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

\((GA\,1)\)   $:$     \(\displaystyle \forall g, h \in G, x \in X:\) \(\displaystyle g * \left({h * x}\right) = \left({g \circ h}\right) * x \)             
\((GA\,2)\)   $:$     \(\displaystyle \forall x \in X:\) \(\displaystyle e * x = x \)             


The group $G$ thus acts on the set $X$.

The group $G$ can be referred to as the group of transformations, or a transformation group.


From Permutation Representation

Let $\operatorname{Sym} \left({X}\right)$ be the group of permutations on $X$.

Let $\rho : G \to \operatorname{Sym} \left({X}\right)$ be a permutation representation.


The group action of $G$ associated to the permutation representation $\rho$ is the group action $\phi : G \times X \to X$ defined by:

$\phi (g, x) = \rho(g)(x)$


Right Group Action

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

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

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

\((RGA\,1)\)   $:$     \(\displaystyle \forall g, h \in G, x \in X:\) \(\displaystyle \left({x * g}\right) * h = x * \left({g \circ h}\right) \)             
\((RGA\,2)\)   $:$     \(\displaystyle \forall x \in X:\) \(\displaystyle x * e = x \)             



Also defined as

A (left) group action is sometimes defined as what on $\mathsf{Pr} \infty \mathsf{fWiki}$ 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

  • Results about group actions can be found here.


Sources