permutation module (awaiting review)

If $H$ is a finite index subgroup of $G$, then $G$ acts on the right cosets of $H$ via $Hg\overset{g'}{\mapsto}Hgg'$, yielding a permutation representation that sends each $g\in G$ to the permutation of $H\backslash G$ given by its action. This defines a homomorphism from $G$ to the symmetric group on $n=[G:H]$ letters.

For a commutative ring $R$, the action of $G$ of $H\backslash G$ can be extended $R$-linearly to an action of $G$ on $R[H\backslash G]$, the free $R$-module with basis $G\backslash H$, which is then a module over the group ring $R[G$]. The $R[G]$-module $R[H\backslash G]$ is a permutation module over $R$.