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 $G$