Permutation module ($R[X]$)

For a finite group action on a finite set $X$, $R[X]$ is the free $R$-module with basis $X$, with the group permuting that basis. For a transitive set $G/H$ it is ${\mathrm{Ind}}_H^{G}R$. Disjoint unions of sets give direct sums of permutation modules.