Let $(H \subset G)$ be an inclusion of finite groups.

Let $\{ g_i \ \vert \ i \in I=[1, \dots ,n] \}$ a subset of $G$ of double coset representatives, i.e. $$G = \coprod_{i \in I} Hg_iH$$
On the algebra $\mathbb{C}G$ we consider the elements $$e_i = \frac{1}{\vert H \vert} \sum_{g \in Hg_iH} g$$
They generate the double coset subalgebra of $\mathbb{C}G$, we have the following constants structure $c_{i j}^{k} \in \mathbb{N}$:
$$e_{i} \cdot e_{j} = \sum_{k \in I} c_{i j}^{k} e_{k}$$
*Remark*: we get the first examples of finite hypergroups (up to a renormalisation), noted $G//H$.

For example, with $(H \subset G) = (\mathbb{Z}/2 \subset \mathbb{Z}/5 \rtimes \mathbb{Z}/2)$ we get the following table (with $g_1 = e$ the neutral):

$$\begin{array}{c|c|c|c|c|c|c}
\cdot & e_1 & e_2 & e_3 \\ \hline
e_1 & e_1 & e_2 & e_{3} \\ \hline
e_2 & e_2 & 2e_1+e_3 & e_2+e_3 \\ \hline
e_{3} & e_{3} & e_{2}+e_3 &2e_1+e_2
\end{array} $$

*Definition*: a basic element $e_{i}$ is called *generating* if $\forall j \in I, \exists r_j$ such that $(e_{i}^{r_j}, e_j) >0$.

*Remark*: It exists a generating basic element iff $\exists g \in G$ such that $\langle H,g \rangle =G$.

It is true for every primitive inclusion (and also for every "distributive" inclusion, see here).

Let the matrix $M_i = (m_{ij}^{(r)})_{rj}$ defined by $e_{i}^{r} = \sum_{j \in I}m_{ij}^{(r)}e_j $.

For the example above: $M_1 = \left(\begin{matrix} 1 & 0 & 0 \\ 1 & 0 & 0 \\ 1 & 0 & 0\end{matrix}\right) $, $M_2 = \left(\begin{matrix} 0 & 1 & 0 \\ 2 & 0 & 1 \\ 0 & 3 & 1\end{matrix}\right)$ and $M_3 = \left(\begin{matrix} 0 & 0 & 1 \\ 2 & 1 & 0 \\ 0 & 1 & 3\end{matrix}\right)$

*Definition*: a basic element $e_{i}$ is called *separating* if the matrix $M_i$ is invertible.

*Remark:* "separating" implies "generating", but the converse is false.

*Definition*: $(H \subset G)$ is called *separating* if it exists a *separating* basic element $e_{i}$.

**Question**: Is a prime index inclusion of finite groups, separating?

*Remark*: It's checked by GAP for $[G:H] \le 500$ and $\vert G \vert \le 4000$.

The first non-separating primitive inclusions are given by:

gap> G:=PrimitiveGroup(d,r);

gap> H:=Stabilizer(G,1);

with $(d,r) = (16,1), (16,2), (25, 1 ),( 25, 4 ),( 25, 6 ),( 25, 11 ), \dots$

For example with $(d,r) = (16,1)$ we get the following table and $M_i$ matrices:

$$\begin{array}{c|c|c|c|c|c|c}
\cdot & e_1 & e_2 & e_3 & e_4 \\ \hline
e_1 & e_1 & e_2 & e_{3} &e_4 \\ \hline
e_2 & e_2 & 5e_1+2e_3+2e_4 & 2e_2+2e_3+e_4 & 2e_2+e_3+2e_4 \\ \hline
e_3 & e_3 & 2e_2+2e_3+e_4 &5e_1+2e_2+2e_4 & e_2+2e_3+2e_4 \\ \hline
e_4 & e_4 & 2e_2+e_3+2e_4 &e_2+2e_3+2e_4 & 5e_1+2e_2+2e_3
\end{array} $$
$M_1 = \left(\begin{smallmatrix} 1 & 0& 0& 0 \\ 1 & 0 & 0 & 0 \\ 1 & 0 & 0 & 0 \\ 1 & 0 & 0 & 0\end{smallmatrix}\right)$, $M_2 = \left(\begin{smallmatrix} 0 & 1& 0& 0 \\ 5 & 0 & 2 & 2 \\ 0 & 13 & 6 & 6 \\ 65 & 24 & 44 & 44\end{smallmatrix}\right)$, $M_3 = \left(\begin{smallmatrix} 0& 0& 1& 0 \\ 5& 2& 0& 2 \\ 0& 6& 13& 6 \\ 65& 44& 24& 44 \end{smallmatrix}\right)$, $M_4 = \left(\begin{smallmatrix} 0& 0& 0& 1 \\ 5& 2& 2& 0 \\ 0& 6& 6& 13 \\ 65& 44& 44& 24\end{smallmatrix}\right)$

*Bonus question*: How to improve the "separating" assumption for becoming true for every primitive inclusion?

Idea to check: widen the notion to the existence of a separating (non-necessarily basic) element.