The p-radical and Hall's theorems for residually thin and π-valenced hypergroups, table algebras, and association schemes

A positive integer n(H) (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. Then 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, Vasil'ev and Zieschang's results on solvable association schemes are generalized. (C) 2022 Elsevier Inc. All rights reserved.