# Definition Let $H$ be a [[subgroup]] of the [[group]] $G$, and $g \in G$. Let $e, h_1, \ldots, h_n \in H$. The assembly $ eg, h_1 g, h_2 g, \ldots, h_n g $ is the *right coset* of $H$. A coset is not a subgroup in general. However, if $g$ is an [[group element|element]] of $H$, then the coset is itself the subgroup $H$ via the [[Group Rearrangement Theorem]]. We can write the cosets more mathematically as: $ \begin{align} gH = \{gh | h \in H\}, g \in G, \\ Hg = \{hg | h \in H\}, g \in G, \end{align} $ where the first definition is for *left cosets* and the second is for *right cosets* # Theorem Two right (or left) cosets of a given subgroup either contain exactly the same elements or have no elements in common, i.e. they are either disjoint or identical as sets.