# Definition(s) > [!NOTE] Definition (Index) > Let $G$ be a [[Groups|group]] and $H$ a [[Subgroup|subgroup]] of $G.$ Then index of $H$ in $G,$ denoted $[G:H]$ is the number of distinct left (or right) [[Coset|cosets]] of $H$ equivalently, $[G:H]$ is the [[Cardinality|cardinality]] of the [[Coset space|left coset space]]. **Note**: We may also denote $[G:H]$ as $|G/H^l|.$ By [[Left and Right Coset Spaces have Same Cardinality]], the number of distinct left cosets is the same as the number of distinct right cosets. > [!Example] Example > $\mathbb{C}^*$ has a subgroup $\mathbb{S}=\{ z\in \mathbb{C}^*: |z|=1 \}.$ > > Then by [[Necessary Condition for Equality of Cosets]], $\alpha \mathbb{S}=\beta \mathbb{S}\iff |\alpha|=|\beta|.$ > > Thus $re^{i\theta}\mathbb{S}=r\mathbb{S}$ hence cosets of $\mathbb{S}$ are circles on the Argand diagram centred at the origin and $[\mathbb{C}^* : \mathbb{S}]=\infty.$ > [!Example] Example > $[D_{8}:R]=2$ where $R$ is the subgroup of rotations # Properties(s) # Application(s) **Examples**: # Bibliography