Primitive bicirculant association schemes and a generalization of Wielandt’s theorem

Abstract:

Bannai and Ito defined association scheme theory as doing “group theory without groups”, thus raising 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 $\mathfrak {X}$ is a primitive bicirculant association scheme of order $2p^e$, $p>2$ is a prime, then $\mathfrak {X}$ is of class at most $2$, and if it is of class exactly $2$, then $2p^e=(2s+1)^2+1$ for some natural number $s$, with the valencies of $\mathfrak {X}$ being $1$, $s(2s+1)$, $(s+1)(2s+1)$, and the multiplicities of $\mathfrak {X}$ being $1$, $p^e$, $p^e-1$. Consequently, translated into permutation group theory language, a primitive permutation group $G$ of degree $2p^e$, $p$ a prime and $e \geq 1$, containing a cyclic subgroup with two orbits of size $p^e$, is either doubly transitive or of rank $3$, in which case $2p^e=(2s+1)^2+1$ 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 $1$, $p^e$ and $p^e-1$.