# $G$-Set

If $G$ is a group and $X$ a set, then $X$ is called a left $G$-Set if there exists a mapping $\lambda :G\times X\to X$ with

$$\lambda ({g}_{1},\lambda ({g}_{2},x))=\lambda ({g}_{1}{g}_{2},x)$$ |

or shorter with $\lambda (g,x)=gx$

$${g}_{1}({g}_{2}(x))=({g}_{1}{g}_{2})(x)$$ |

for all $x\in X$ and ${g}_{1},{g}_{2}\in G$. And when $G$ acts on a set $X$, the set $X$ is always a $G$-set.

$X$ is called a right $G$-Set if there exists a mapping $\lambda :X\times G\to X$ with

$$\lambda (\lambda (x,{g}_{2}),{g}_{1})=\lambda (x,{g}_{2}{g}_{1})$$ |

Title | $G$-Set |
---|---|

Canonical name | GSet |

Date of creation | 2013-03-22 17:55:44 |

Last modified on | 2013-03-22 17:55:44 |

Owner | jwaixs (18148) |

Last modified by | jwaixs (18148) |

Numerical id | 5 |

Author | jwaixs (18148) |

Entry type | Definition |

Classification | msc 20-00 |

Related topic | GroupAction |