# group action

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

1. 1.

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

2. 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. 1.

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

2. 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 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 2013-03-22 12:12:17 Last modified on 2013-03-22 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