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 $G$.