Definition:Group Action

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This page is about Group Action. For other uses, see Action.

Definition

Let $X$ be a set.

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


Left Group Action

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:

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


Right Group Action

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:

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


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 $\struct {\map \Gamma X, \circ}$ be the symmetric group on $X$.

Let $\rho: G \to \struct {\map \Gamma X, \circ}$ 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:

$\map \phi {g, x} = \map {\rho_g} x$

where $\rho_g : X \to X$ is the permutation representation associated to $\rho$ for $g \in G$ by $\map {\rho_g} x = \map \phi {g, x}$.


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 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 refer to a group action as a $G$-action, and refer to the set $X$ upon which it acts as 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 $\map {\phi_g} x$ 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.


Examples

Cyclic Group on Polygon

Consider the cyclic group $C_n$ defined as $\gen g$ whose identity is $e$.

Let $P_n$ be a regular $n$-sided polygon.


Then $C_n$ acts on on $P_n$ by the mapping for which, for each vertex $x$ of $P_n$:

$e x = x$
$g^k x$ is the vertex obtained when $P_n$ is rotated through $\dfrac {2 \pi k} n$ radians about the center of $P_n$.


Also see

  • Results about group actions can be found here.


Sources

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