Primitive Permutation Groups of Simple Diagonal Type

Let G be a finite primitive group such that there is only one minimal normal subgroup M in G, this M is nonabelian and nonsimple, and a maximal normal subgroup of M is regular. Further, let H be a point stabilizer in G. Then H (1 M is a (nonabelian simple) common complement in M to all the maximal normal subgroups of M, and there is a natural identification ofMwith a direct power T m of a nonabelian simple group T in which H ~ M becomes the "diagonal" subgroup of T m: this is the origin of the title. It is proved here that two abstractly isomorphic primitive groups of this type are permutationally isomorphic if (and obviously only if) their point stabilizers are abstractly isomorphic. Given T m, consider first the set of all permutational isomorphism classes of those primitive groups of this type whose minimal normal subgroups are abstractly isomorphic to T m. Secondly, form the direct product S,~ × Out Tof the symmetric group of degree m and the outer automorphism group of T(so Out T = Aut T/Inn T), and consider the set of the conjugacy classes of those subgroups in S,~ X Out Twhose projections in S,~ are primitive. The second result of the paper is that there is a bijection between these two sets. The third issue discussed concerns the number of distinct permutational isomorphism classes of groups of this type, which can fall into a single abstract isomorphism class. The a im o f this pape r is to invest igate p r imi t ive p e r m u t a t i o n groups G on finite sets fl , such that G has a un ique m i n i ma l no rma l subgroup M , this M is nonabe l ian and nonsimple , and a m a x i m a l no rma l subgroup K o f M is regular. Such groups a re -somet imes called primitive groups o f simple diagonal type. Let H be a poin t stabil izer in such a G. Then H N M i s a (nonabel ian simple) c o m m o n c o m p l e m e n t in M to all the m a x i m a l no rma l subgroups o f M , and there is a natural way o f identifying M w i t h a direct power T m o f a nonabe l ian s imple group T in such a way tha t H M M becomes the diagonal diag T m. [Let K~ . . . . , K~, be the m ax i m a l no rma l subgroups o f M . To each e lement x o f M , Received September 3, 1987