# Definition:Group Action

## Contents

## Definition

Let $X$ be a set.

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

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

- $\forall \tuple {g, x} \in G \times X: g * x := \map \phi {\tuple {g, x} } \in X$

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

\((GA \, 1)\) | $:$ | \(\displaystyle \forall g, h \in G, x \in X:\) | \(\displaystyle g * \paren {h * x} = \paren {g \circ h} * x \) | |||||

\((GA \,2)\) | $:$ | \(\displaystyle \forall x \in X:\) | \(\displaystyle e * x = x \) |

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 $\struct {\map \Gamma X, \circ}$ 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 $\map {\rho_g} x = \map \phi {g, x}$.

### 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 $\map {\phi_g} x$ 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.

## Also see

- Results about
**group actions**can be found here.

## Sources

- 1965: J.A. Green:
*Sets and Groups*... (previous) ... (next): $\S 5.5$ - 1971: Allan Clark:
*Elements of Abstract Algebra*... (previous) ... (next): Chapter $2$: The Sylow Theorems: $\S 53$ - 1982: P.M. Cohn:
*Algebra Volume 1*(2nd ed.) ... (previous) ... (next): $\S 3.3$: Group actions and coset decompositions - 1996: John F. Humphreys:
*A Course in Group Theory*... (previous) ... (next): $\S 10$: Definition $10.1$