# coset

Let $H$ be a subgroup^{} of a group $G$, and let $a\in G$. The left coset^{} of $a$ with respect to $H$ in $G$ is defined to be the set

$$aH:=\{ah\mid h\in H\}.$$ |

The right coset of $a$ with respect to $H$ in $G$ is defined to be the set

$$Ha:=\{ha\mid h\in H\}.$$ |

Two left cosets $aH$ and $bH$ of $H$ in $G$ are either identical or disjoint. Indeed, if $c\in aH\cap bH$, then $c=a{h}_{1}$ and $c=b{h}_{2}$ for some ${h}_{1},{h}_{2}\in H$, whence ${b}^{-1}a={h}_{2}{h}_{1}^{-1}\in H$. But then, given any $ah\in aH$, we have $ah=(b{b}^{-1})ah=b({b}^{-1}a)h\in bH$, so $aH\subset bH$, and similarly $bH\subset aH$. Therefore $aH=bH$.

Similarly, any two right cosets $Ha$ and $Hb$ of $H$ in $G$ are either identical or disjoint. Accordingly, the collection^{} of left cosets (or right cosets) partitions^{} the group $G$; the corresponding equivalence relation^{} for left cosets can be described succintly by the relation^{} $a\sim b$ if ${a}^{-1}b\in H$, and for right cosets by $a\sim b$ if $a{b}^{-1}\in H$.

The index of $H$ in $G$, denoted $[G:H]$, is the cardinality of the set $G/H$ of left cosets of $H$ in $G$.

Title | coset |
---|---|

Canonical name | Coset |

Date of creation | 2013-03-22 12:08:10 |

Last modified on | 2013-03-22 12:08:10 |

Owner | djao (24) |

Last modified by | djao (24) |

Numerical id | 9 |

Author | djao (24) |

Entry type | Definition |

Classification | msc 20A05 |

Defines | index |

Defines | left coset |

Defines | right coset |