Solvable groups derived from hypergroups. (Q2797022)

In this paper the authors extend their previous results related to nilpotent groups derived from hypergroups [J. Algebra 382, 177-184 (2013; Zbl 1286.20079)] to solvable groups by slight modification of the relation \(\nu\). -- Indeed, for any hypergroup \(H\) and any natural number \(n\), they define a reflexive and symmetric relation \(\tau_n\) by setting first \(H^{(0)}=H\) and \(H^{(k+1)}=\{h\in H^{(k)}\mid x\cdot y\cap h\cdot y\cdot x\neq\emptyset\text{ for some }x,y\in H^{(k)}\}\), for all \(k\geq 0\), and then putting \(\tau_n=\bigcup_{m\geq 1}\tau_{m,n}\), where \(\tau_{1,n}=\bigtriangleup_H\) is the diagonal relation on \(H\) and for every integer \(m>1\), \(\tau_{m,n}\) is the relation defined by: \(x\tau_{m,n}y\Leftrightarrow\exists z_1,\ldots,z_m\in H,\;\sigma\in S_n:\sigma(i)=i\text{ if }z_i\not\in H^{(n)}\text{ such that }x\in\prod_{i=1}^m z_i\text{ and }y\in\prod_{i=1}^mz_{\sigma(i)}\).NEWLINENEWLINE They show that the relation \(\tau_n^*\) is a strongly regular relation. Moreover, it is proved that the relation \(\tau^*=\bigcap_{n\in\mathbb N}\tau_n^*\) is the smallest strongly regular relation on a finite hypergroup \(H\) such that \(H/\tau^*\) is a solvable group. Finally, in the last section of the paper, they prove that the set \(p^{-1}(1)\), called \(\tau\)-heart of \(H\), is also a \(\tau\)-part of \(H\), where \(p\colon H\to H/\tau\) is the canonical projection.