A Problem of Wielandt on Finite Permutation Groups

Problem 6.6 in the Kourovka Notebook [9], posed by H. Wielandt, reads as follows. ' Let P, Q be permutation representations of a finite group G with the same character. Suppose P(G) is a primitive permutation group. Is Q{G) necessarily primitive? Equivalently: Let A, B be subgroups of a finite group G such that for each class C of conjugate elements of G their intersection with C has the same cardinality: \A n C\ = \B n C\. Suppose A is a maximal subgroup of G. Is B necessarily maximal? The answer is known to be affirmative if G is soluble.' Note that two permutation representations with the same character have the same kernel, and that one may as well consider the question as one concerning the factor group over that common kernel; we may therefore restrict attention to faithful representations. Our aim is to reduce the restricted question to the case of almost simple G (we call a group almost simple if its socle is non-abelian and simple). The solution of the problem may then be approached by examining permutation representations of the almost simple groups. While much progress has been made lately towards understanding maximal subgroups, and therefore primitive permutation representations, of almost simple groups, here one would need to know also many non-maximal subgroups; thus we are far from having reduced the issue to scanning existing tabulations. Still it seems plausible that the almost simple case can eventually be settled and that for that case the answer will be affirmative—at worst, with an explicit list of exceptions. Our reduction is reversible in the sense that it will convert such a result into a general theorem by showing how to generate the full list of exceptions from that of the almost simple ones. (If there are no exceptions to list, restriction to the faithful case will have been immaterial; but if there are any exceptions at all, the non-faithful ones are clearly beyond accounting, as the hypothesis places no restriction whatsoever on the common kernel.)