Primitive bicirculant association schemes and a generalization of Wielandt's theorem
Bannai and Ito defined association scheme theory as doing “group theory without groups”, thus rasing a basic question as to which results about permutation groups are, in fact, results about association schemes? By considering transitive permutation groups in a wider setting of association schemes, it is shown in this paper that one such result is the classical theorem of Wielandt about primitive permutation groups of degree 2p, p a prime, being of rank at most 3 (see Math. Z. 63 (1956), 478–485). More precisely, it is proved here that if X is a primitive bicirculant association scheme of order 2pe, p > 2 is a prime, then X is of class at most 2, and if it is of class exactly 2, then 2pe = (2s + 1)2 + 1 for some natural number s, with the valencies of X being 1, s(2s+1), (s+1)(2s+1), and the multiplicities of X being 1, pe, pe−1. Consequently, translated into permutation group theory language, a primitive permutation group G of degree 2pe, p a prime and e 1, containing a cyclic subgroup with two orbits of size pe, is either doubly transitive or of rank 3, in which case 2pe = (2s + 1)2 + 1 for some natural number s, the sizes of suborbits of G are 1, s(2s + 1) and (s + 1)(2s + 1), and the degrees of the irreducible constituents of G are 1, pe and pe − 1.