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# group action

Let $G$ be a group and let $X$ be a set. A left group action is a function $\cdot:G\times X\longrightarrow X$ such that:

1. $1_{G}\cdot x=x$ for all $x\in X$

2. $(g_{1}g_{2})\cdot x=g_{1}\cdot(g_{2}\cdot x)$ for all $g_{1},g_{2}\in G$ and $x\in X$

A right group action is a function $\cdot:X\times G\longrightarrow X$ such that:

1. $x\cdot 1_{G}=x$ for all $x\in X$

2. $x\cdot(g_{1}g_{2})=(x\cdot g_{1})\cdot g_{2}$ for all $g_{1},g_{2}\in G$ and $x\in X$

There is a correspondence between left actions and right actions, given by associating the right action $x\cdot g$ with the left action $g\cdot x:=x\cdot g^{{-1}}$. In many (but not all) contexts, it is useful to identify right actions with their corresponding left actions, and speak only of left actions.

Special types of group actions

A left action is said to be effective, or faithful, if the function $x\mapsto g\cdot x$ is the identity function on $X$ only when $g=1_{G}$.

A left action is said to be transitive if, for every $x_{1},x_{2}\in X$, there exists a group element $g\in G$ such that $g\cdot x_{1}=x_{2}$.

A left action is free if, for every $x\in X$, the only element of $G$ that stabilizes $x$ is the identity; that is, $g\cdot x=x$ implies $g=1_{G}$.

Faithful, transitive, and free right actions are defined similarly.

## Mathematics Subject Classification

16W22*no label found*20M30

*no label found*

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## Comments

## Monoid actions

On PlanetMath are defined group actions.

In the book "Abstract and Concrete Categories" (http://katmat.math.uni-bremen.de/acc/acc.pdf) I found the definition of monoid actions. Are these a generalization of group actions? Should "monoid action" be added to the encyclopedia?

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Victor Porton - http://www.mathematics21.org

* Algebraic General Topology and Math Synthesis