# Definition:Group Action

*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

- 1989: Ephraim J. Borowski and Jonathan M. Borwein:
*Dictionary of Mathematics*... (previous) ... (next):**act**(of a group) - 2008: David Nelson:
*The Penguin Dictionary of Mathematics*(4th ed.) ... (previous) ... (next):**action**:**3.**(of a group on a set) - 2021: Richard Earl and James Nicholson:
*The Concise Oxford Dictionary of Mathematics*(6th ed.) ... (previous) ... (next):**action, acts**