## 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 X is a primitive bicirculant association scheme of order 2p^{e}, p > 2 is a prime, then 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 X being 1, s(2s + 1), (s + 1)(2s + 1), and the multiplicities of 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 ≥ 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.

Original language | English |
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Pages (from-to) | 3203-3221 |

Number of pages | 19 |

Journal | Transactions of the American Mathematical Society |

Volume | 362 |

Issue number | 6 |

DOIs | |

State | Published - 1 Jun 2010 |

Externally published | Yes |