Generalization of group rings to quotient groups The structure of a group ring $R[G]$, where $G$ is a group and $R$ is a ring is generalized to monoid rings $R[M]$, and also semigroup rings.
Is there such a things as a "quotient group ring" $R[G/H]$? Obviously it should be a quotient group $G/H$ with $H$ not necessarily normal, or it would be the same as a group ring. It should also be both a free module and a ring.
 A: I think describing the induced representation $\text{Ind}_H^G 1$ may answer the question. Let $G$ be a finite group.


*

*Think to $\mathbb{C}[G]$ as a ring of linear maps, generated by $\{\pi(g),g \in G\}$, where $\pi(g)$ is the linear map $\mathbb{C}^{|G|} \to \mathbb{C}^{|G|}$ permuting the indices following the group law : $\forall f \in \mathbb{C}^{|G|}, \ \ \pi(g) f(g') = f(gg')$.

*$H$ is a subgroup. There are some $g_j \in G$ such that $G = \bigcup_{j=1}^{|G/H|} g_j H$ where the union is disjoint. 
For any $g \in G$ and $j$, there is some $i= i(g,j)$ such that $g g_j\in g_i H$, ie. for some $h\in H$, $g g_j = g_i h$ and $g g_i H = g_j H$.
Let $\rho(g)$ be the corresponding linear map $\mathbb{C}^{|G/H|} \to \mathbb{C}^{|G/H|}$ permuting the indices this way : $\rho(g) f(g_j H) = f(g g_j H) = f(g_{i(g,j)} H)$.
The $\mathbb{C}$ linear combinations of the $\rho(g)$ is a ring, call it $\mathbb{C}[G/H]$.
In other words $\mathbb{C}[G/H]$ is a ring of linear maps acting on the vector space of right $H$-invariant functions $|G| \to \mathbb{C}$ (this formulation should work for non-finite groups)

*Thus $\rho(h), h  \in H$ isn't the identity, since it acts by left multiplication on those right $H$-invariant functions.
Except of course when $H$ is normal, so that $g_j H = H g_j$ and right $H$-invariant implies left $H$-invariant, in which case $\mathbb{C}[G/H]$ is also the group ring of the group $G/H$.

*In term of $G$ modules ?  In general $\mathbb{C}[G/H]$ is a quotient of $\mathbb{C}[G]$.
