The Affine Permutation Groups of Rank Three

(iii) G is an ajfine group, that is, the socle of G is a vector space V, where V s= {ZpY for some prime p and n=p ; moreover, if Go is the stabilizer of the zero vector in V then G = VG0, Go is an irreducible subgroup of GLd(p), and GQ has exactly two orbits on the non-zero vectors of V. The rank 3 groups under (i) are given by the classification of the 2-transitive groups with simple socle (see § 5 of [5]). Those under (ii) are determined in [22] when the socle socG is a classical group, in [3] when socG is an alternating group, and in [27] when soc G is an exceptional group of Lie type or a sporadic group. The object of this paper is to determine completely the groups satisfying (iii), thus completing the classification of the finite primitive groups of rank 3. Our proof uses the classification of finite simple groups. The 2-transitive affine groups have been determined by Hering in [15]. Our methods can easily be used to give a proof of Hering's result, and we present this proof in Appendix 1, at the end of the paper. The finite soluble primitive groups of rank 3 are automatically affine groups, and these are determined by Foulser in [11]. We shall make use of his results and of some of his methods in one special case in our proof, which we call the 'extraspecial case'. Our main result is