group action
Let $G$ be a group and let $X$ be a set. A left group action^{} is a function $\cdot :G\times X\u27f6X$ 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 $\cdot :X\times G\u27f6X$ 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.
Title  group action 
Canonical name  GroupAction 
Date of creation  20130322 12:12:17 
Last modified on  20130322 12:12:17 
Owner  djao (24) 
Last modified by  djao (24) 
Numerical id  10 
Author  djao (24) 
Entry type  Definition 
Classification  msc 16W22 
Classification  msc 20M30 
Related topic  Group 
Defines  effective 
Defines  effective group action 
Defines  faithful 
Defines  faithful group action 
Defines  transitive 
Defines  transitive group action 
Defines  left action 
Defines  right action 
Defines  faithfully 
Defines  action 
Defines  act on 
Defines  acts on 