**Update:** Originally, I formulated this question for finite abelian hypergroups, but in a discussion with Geoff Robinson below I realized that the abelian hypergroups defined below are equivalent to *table algebras* up to a simple normalization [reference]. Hence, I am equally interested on the table algebra case.

## Question

A collaborator and I are investigating finite abelian hypergroups (see a short definition below or the following surveys [survey1],[survey2], in a quantum computing context. Our interest is more focused on finite hypergroups that arise from finite groups (namey, conjugacy class and character hypergroups) but we would also be interested on knowing about the arbitrary finite abelian hypergroup case. The question is:

Question.Is there a classification of finite conjugacy class / character hypergroups? If not, is this problem considered to be "impossible" (or, more precisely, wild)?

Of course, this question is very easy for finite abelian groups. Yet, the case for finite abelian hypergroups seems actually quite *hard*: as formulated, our question is probably equivalent to asking whether classifying finite groups "up to conjugation" and their characters is believed to be as hard as the problem of classifying finite groups. The latter problem, in turn, is wild.

Still, I have not rigorous way to turn this intuition into an argument. My belief is not fully rigorous because classifying groups "up to conjugation" and their characters without classifying groups could (in principle) be a lot easier than classifying groups. So, can something more rigorous be said?

Definition.We are using the following standard definition: a finite abelian hypergroup is a set $H = \{ C_0 , C_1 , . . . , C_n\}$ together with an associative unital abelian algebra structure on $\mathbb{R}H$ $$C_iC_j=\sum_{k}n_{ij}^{k}C_k$$and an involution $*: H \rightarrow H$ such that

- $n_{ij}^{k}\geq 0$
- $\sum_{k} n_{ij}^{k}=1$
- $C_0$ is the identity
- $n_{ij}^{0}> 0$ if and only if $C_i^*=C_j$

**Example 1.** The irreducible characters $\widehat{G}$ of any finite group $G$ define a hypergroup using the tensor product of a characters the multiplication of the algebra and the complex conjugation as involution. This follows from the formula $$\chi_\mu \otimes \chi_\nu = \sum_{\gamma\in\widehat{G}} m_{\mu,\nu}^\gamma \chi_\gamma.$$
The coefficients $m_{\mu\nu}^\gamma$ in that expansion are not normalized, but one can easily normalize characters so that condition 1 is met [2].

**Example 2.** The conjugacy classes of any finite group $G$ also have a natural hypergroup structure (this is what we call conjugacy class hypergroup). The simplest way to define this hypergroup is to identify every class $C_g\subset G$ witht he following element of the group algebra $\mathbb{R}G$:

$$C_g = \frac{1}{|C_g|}\sum_{aga^{-1}\in C_g} aga^{-1}$$.

It is shown in [1] that the classes with the product in $\mathbb{R}G$ define a hypergroup.