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# 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 $c=ah_{1}$ and $c=bh_{2}$ 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 $ah=(bb^{{-1}})ah=b(b^{{-1}}a)h\in bH$, 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 $ab^{{-1}}\in H$.

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

## Mathematics Subject Classification

20A05*no label found*

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