4 1. INTRODUCTION For these reasons we have considered it worth the effort to obtain reasonably complete reduction or classification results for (arithmetically) exceptional permu- tation groups. For our original question about the analog of Schur's question, we could have used the genus 0 condition on G (see Section 2.6) at earlier stages to remove many groups from consideration before studying exceptionality. A special case of exceptionality has arisen recently in some work on graphs (see [19] and [36]). The question there involves investigating those exceptional groups in which the quotient A/G is cyclic of prime power order. If G A are transitive permutation groups on n points, with U A the stabilizer of a point, then A is primitive if and only if U is a maximal subgroup of A. Suppose that A is imprimitive, so there is a group M with U M A. If (A, G) is arithmetically exceptional, then so are the pairs (M, MOG) (in the action on M/U) and (A, G) in the action on A/M. The converse need not hold (but it holds if A/G is cyclic, see Lemma 3.5). So our main focus is on primitive permutation groups. We use the Aschbacher- O'Nan-Scott classification of primitive groups, which divides the primitive permu- tation groups into five classes (cf. Section 3.2). The investigation of these five classes requires quite different arguments. Exceptionality arises quite often in affine per- mutation groups, one cannot expect to obtain a reasonable classification result. Let (A, G) be arithmetically exceptional, and A primitive but not affine. We show that A does not have a regular nonabelian normal subgroup, which com- pletely removes one of the five types of the Aschbacher-0 'Nan-Scott classification. If A preserves a product structure (that means A is a subgroup of a wreath prod- uct in product action), then we can reduce to the almost simple case. If A/G is cyclic, A does not preserve a product structure, but acts in diagonal action, then we essentially classify the possibilities (up to the question of existence of outer au- tomorphisms of simple groups with a certain technical condition). So the main case to investigate is the almost simple action. We obtain the following classification, where all the listed cases indeed do give examples. All these examples are groups of Lie type. There are many possibilities if the Lie rank is one. For higher Lie rank, the stabilizers are suitable subfield subgroups. THEOREM 1.5. Let A be a primitive permutation group of almost simple type, so L A Aut(L) with L a simple nonabelian group. Suppose there are subgroups B and G of A with G A and B/G cyclic, such that (JE?, G) is exceptional. Let M be a point stabilizer in A. Then L is of Lie type, and one of the following holds. (a) L = PSL2(2a), G = L and either (i) M n L = PSL2(26) with 3 a/b r a prime and B/G generated by a field automorphism x such that \CL(X)\ is not divisible by r or (ii) a 1 is odd and M C\G is dihedral of order 2(q + 1). (b) L = PSL2(g) with q = pa odd, G = L or PGL2(g) and either (i) M fl L = PSL2O?6) with a/b r a prime and r not a divisor of p(p2 1), and B any subgroup containing an automorphism x which is either a field automorphism of order e or the product of a field automorphism of order e and a diagonal automorphism, such that r does not divide p2a/e 1 or (ii) MnL = L2(pa/p) andB is a subgroup of Aut(G) such thatp\[B : G] and B contains a diagonal automorphism or
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