Primitive permutation groups with a regular subgroup

This paper starts the classification of the primitive permutation groups (G, Q) such that G contains a regular subgroup X. We deter-mine all the triples (G, Omega, X) with soc(G) an alternating, or a sporadic or an exceptional group of Lie type. Further, we construct all the examples (G, Omega, X) with G a classical group which are known to us. Our particular interest is in the 8-dimensional orthogonal groups of Witt index 4. We determine all the triples (G, Omega, X) with soc(G) congruent to P Omega(+)(8) (q). In order to obtain all these triples, we also study the almost simple groups G with G congruent to P Omega 2(n+1) (q). The case G congruent to U-n(q) is started in this paper and finished in [B. Baumeister, Primitive permutation groups of unitary type with a regular subgroup, Bull. Belg. Math. Soc. 112 (5) (2006) 657-673]. A group X is called a Burnside-group (or short a B-group) if each primitive permutation group which contains a regular subgroup isomorphic to X is necessarily 2-transitive. In the end of the paper we discuss B-groups. (c) 2006 Elsevier Inc. All rights reserved.