The \(\pi \)-radical and Hall's theorems for residually thin and \(\pi \)-valenced hypergroups, table algebras, and association schemes (Q2118944)

A positive integer \(nH\) (the valency) is assigned to any residually thin finite hypergroup \(H\); and for any set \(\pi\) of prime numbers, \(\pi\)-valency is defined for elements of \(H\). These notions coincide with those of valency (resp. order and degree) when $H$ appears as the relations of an association scheme (resp. the distinguished basis of a standard table algebra). When all elements of $H$ are \(\pi\)-valenced, existence is established of a closed subset $U$ (called the \(\pi\)-radical of \(H\)) such that $nU$ is a \(\pi\)-number, $U$ contains all subnormal closed subsets $V$ with $nV$ a \(\pi\)-number, and the quotient hypergroup $H//U$ is a group. The classical results on existence and conjugacy of Hall $\pi$-subgroups for $H//U$ when the group is solvable (or more generally \(\pi\)-separable), and Sylow theory for the group when \(\pi\) is a singleton set, are extended to the Hall \(\pi\)-subsets of the hypergroup \(H\). Thus, \textit{A. V. Vasil'ev} and \textit{P.-H. Zieschang}'s results on solvable association schemes are generalized [J. Algebra 594, 733--750 (2022; Zbl 1491.20175)].