# Definition(s) > [!NOTE] Definition (Left & Right Coset Spaces) > Let $G$ be a [[Group|group]]. Let $H$ be a [[Subgroup|subgroup]] of $G.$ > > The left coset space (of $G$ modulo $H$), denoted $G/H^{l}$ is the set of all the left [[Coset|cosets]] of $H$ in $G.$ > > The right coset space (of $G$ modulo $H$), denoted $G/H^{r}$ is the set of all the right cosets of $H$ in $G.$ **Note**: If $H$ is a [[Normal Subgroup|normal subgroup]] of $G$ then $G/H^{l}=G/H^{r}=G/H.$ > [!Example] Examples (In general, left coset space does not equal right coset space modulo same subgroup) > Let $S_{3}=\{ \text{Id}, (1,2), (1,3),(2,3), (1,2,3),(1,3,2) \}$ and $H=\{ 1,(1,2) \}.$ Then $S_{3}/H^l = \{ \{ \text{Id}, (1,2) \}, \{ (1,3), (1,2,3) \}, \{ (2,3), (1,3,2) \} \}$while$S_{3}/H^r = \{ \{ \text{Id}, (1,2) \}, \{ (1,3), (1,3,2) \}, \{ (2,3), (1,2,3) \} \}.$ > [!NOTE] Definition (In terms of equivalence relation) > The left coset space (of $G$ modulo $H$), denoted $G/H^{l}$ is the [[Quotient Set|quotient set]] of $G$ by [[Congruence Modulo Subgroup|left congruence modulo H]]. > > The right coset space (of $G$ modulo $H$), denoted $G/H^{r}$ is the [[Quotient Set|quotient set]] of $G$ by [[Congruence Modulo Subgroup|right congruence modulo H]]. # Properties By [[Coset Space Partitions Subgroup]], ... # Applications See [[Quotient Group]] and [[Quotient Ring]].