Some simplifications in the proof of the Sims conjecture
aa r X i v : . [ m a t h . G R ] F e b Some simplifications in the proof of the Sims conjecture
L´aszl´o Pyber ∗ and Gareth Tracey † Alfr´ed R´enyi Institute of Mathematics, Hungarian Academy of Sciences, Re´altanodautca 13-15, H-1053, Budapest, Hungary
February 15, 2021
Abstract
We prove an elementary lemma concerning primitive amalgams and use it to greatly simplifythe proof of the Sims conjecture in the case of almost simple groups.
Over the last forty years or so, the Classification of Finite Simple Groups (henceforth abbrevi-ated to CFSG) has been greatly utilized to resolve a number of open questions and conjecturesin finite group theory. One of the most well-known of these is the
Sims conjecture , first proposedby Charles Sims in [8]. The conjecture states that if G is a primitive permutation group, and h > H in G , then H has order boundedabove by a function of h .This conjecture was proved by P.J. Cameron, C.E. Praeger, J. Saxl, and G.M. Seitz in [1].Therefore, it is now a theorem, and reads precisely as follows. Theorem 1.1.
There exists a function f : Z → R such that whenever G is a primitive permu-tation group, and h > is the length of a non-trivial orbit of a point stabilizer H in G , then | H | ≤ f ( h ) . In this note, we present a simplified proof of Theorem 1.1. Our proof does still require theCFSG, but the more involved methods used in [1, page 502 and Section 4] are not required.As a by-product of our approach, we also make progress on a more general version of the Simsconjecture, which we will refer to as the non-geometric Goldschmidt-Sims conjecture , followingGoldschmidt’s work in [5]. ∗ [email protected] † [email protected] work of the authors on the project leading to this application has received funding from the EuropeanResearch Council (ERC) under the European Unions Horizon 2020 research and innovation programme (grantagreement No. 741420).
1o state this, we first need to introduce the language of primitive amalgams . A triple offinite groups (
H, M, K ), usually written
H > M < K , is called a primitive amalgam if M is asubgroup of both H and K and:(i) Whenever A is a normal subgroup of H contained in M , we have N K ( A ) = M ; and(ii) whenever B is a normal subgroup of K contained in M , we have N H ( B ) = M .The non-geometric Goldschmidt-Sims conjecture reads as follows: The non-geometric Goldschmidt-Sims conjecture.
There exists a function f : Z × Z → R such that whenever H > M < K is a primitive amalgam with | H : M | = h and | K : M | = k ,then | M | ≤ f ( h, k ) . We remark that the conjecture as stated above is referred to as the ‘baby Goldschmidt-Sims problem’ by Fan in [4]. The general ‘Goldschmidt-Sims problem’ seeks to classify allprimitive amalgams, usually under certain additional conditions (for example, the case where M is a p -group and | H : M | , | K : M | are both prime and distinct from p is dealt with in [4]).We also remark that if ( H, M, K ) is a primitive amalgam, then there is an associated graphΓ = Γ(
H, M, K ) which encodes many properties of the amalgam. The ‘geometric Goldschmidt-Sims conjecture’ is then a statement about this graph. See [3, 5] for more details.Notice that if G is a primitive permutation group, and G α and G β are point stabilizers in G ,then G α > G α ∩ G β < G β is a primitive amalgam. Thus, the non-geometric Goldschmidt-Simsconjecture really is a generalization of the Sims conjecture.A number of papers (see [2, 3, 6, 10, 12, 13, 11]) have addressed and/or made progress on thenon-geometric Goldschmidt-Sims conjecture. In particular a result of of Wielandt (publishedby Knapp in [6, Theorem 2.1]), extending an earlier result of Thompson [10, Main Theorem],implies the so-called Thompson–Wielandt theorem (see Theorem 2.1 below) which asserts thatif
H > M < K is a primitive amalgam, then there exists a prime p such that | H : O p ( H ) | canbe bounded in terms of | H : M | and | K : M | . We remark that [6, Theorem 2.1] is only statedin the case where H and K are maximal subgroups in a finite group G , and M := H ∩ K iscore-free in G . It can be readily seen, however, that the proof works in the more general settingof primitive amalgams stated in Theorem 2.1 below.Our contribution to the non-geometric Goldschmidt-Sims conjecture reads as follows. Theorem 1.2.
There exists a function f : Z × Z → R such that whenever H > M < K isa primitive amalgam with | H : M | = h and | K : M | = k , and either H or K is abelian by ( h, k ) -bounded, then | H | ≤ f ( h, k ) and | K | ≤ f ( h, k ) . We remark that in Theorem 1.2, and throughout the paper, the terminology X is ( n , . . . , n r ) -bounded , for natural numbers X, n , . . . , n r , means that X can be bounded by a function of n , . . . , n r . All other notation used in the paper is standard, though for a finite simple group X of Lie type, we will use Inndiag( X ) to denote the subgroup of Aut( X ) generated by the groupInn( X ) of inner automorphisms of X , together with the set of diagonal automorphisms of X .That is, Inndiag( X ) = h Inn( X ) , α : α a diagonal automorphism of X i .2 Proofs of Theorems 1.1 and 1.2
We begin preparations toward the proof of Theorem 1.2, and the simplified proof of Theorem1.1, with the Thompson–Wielandt theorem [6, 10] mentioned in the introduction.
Theorem 2.1.
Let
H > M < K be a primitive amalgam, with | H : M | = h and | K : M | = k .Then there exists a prime p such that p , | H : O p ( H ) | , and | K : O p ( K ) | are all ( h, k ) -bounded. We remark that the proof of the Thompson–Wielandt theorem does not require the CFSG.We now discuss the proof of the Sims conjecture in [1]. The proof has five steps:(1) Reduction to the case where the primitive group G in question is almost simple with socle X a finite group of Lie type.(2) Proof that it suffices to bound | O p ( H ) ∩ Inndiag( X ) | in terms of h , where p is as in theThompson–Wielandt theorem.(3) Proof that | H | is h -bounded if X has characteristic p .(4) Proof that | H | is h -bounded if X is a classical group in p ′ characteristic.(5) Proof that | H | is h -bounded if X is a finite group of Lie type in p ′ characteristic, and of h -bounded (Lie) rank.Steps (1), (3), and (4) make use of classical results in group and representation theory, and thetheory of algebraic groups. We will not repeat the details here, and instead refer the readerto [1, Section 1 and first three paragraphs of Section 2] for Step (1); the two paragraphs in [1,Section 3] for Step (3); and [1, Section 4a] for Step (4).Steps (2) and (5) are much more involved, and rely on delicate results of Seitz [7] from thetheory of algebraic groups (see page 502 and Section 4 of [1] for the details). For this reason,our aim in this note is to replace the proofs in Steps (2) and (5) with much simpler arguments.Our next result, part (i) of which is Theorem 1.2, will allow us to do so. We note thatwithin its proof, we frequently use the standard fact that if A , B , and C are subgroups of agroup G with C a finite index subgroup of B , then | A ∩ B : A ∩ C | ≤ | B : C | . We will also usethe notation J ( X ) for the Thompson subgroup of the finite group X . That is, J ( X ) := h A Let H > M < K be a primitive amalgam with | H : M | = h and | K : M | = k .(i) Suppose that H is abelian by ( h, k ) -bounded. Then | H | , and hence | K | , is ( h, k ) -bounded.(ii) Suppose that H and K are subgroups of a finite group G , and that I is a normal subgroupof G . If H ∩ I and K ∩ I are abelian by ( h, k ) -bounded, then the exponents of H ∩ I and K ∩ I are ( h, k ) -bounded. roof. Adopt the notation of the statement of the theorem, and set N := H , L := K , in case(i), and N := H ∩ I , L := K ∩ I , in case (ii).Let A be an abelian subgroup of N of maximal order. Then | N : A | is ( h, k )-bounded, since N is abelian by ( h, k )-bounded. Suppose now that K ∩ A is core-free in K . Then K is a transitivepermutation group of degree | K : K ∩ A | = | K : K ∩ N || K ∩ N : K ∩ A | ≤ | K : K ∩ N || N : A | . If weare in case (i), then N = H and | K | can be bounded in terms of | K : K ∩ H || H : A | ≤ k | H : A | .Thus, | K | , and hence | H | , is ( h, k )-bounded. If we are in case (ii), then core K ( L ∩ A ) = 1.Thus, K is a permutation group of degree | K : L ∩ A | . The groups induced by the normalsubgroup L on each of its orbits are all permutation groups of degree | L : L ∩ A | , and are allisomorphic to each other. Hence, the exponent of L is | L : L ∩ A | -bounded. But | L : L ∩ A | = | L : L ∩ N || L ∩ N : L ∩ A | = | I ∩ K : I ∩ M || L ∩ N : L ∩ A | ≤ | K : M || N : A | = k | N : A | . Thus, L has ( h, k )-bounded exponent. Since | N : N ∩ L | = | I ∩ H : I ∩ M | ≤ h , it follows that N has( h, k )-bounded exponent.Thus, we may assume that C := core K ( K ∩ A ) is non-trivial. Since A is abelian and C ≤ A , we have A ≤ N H ( C ), and since the amalgam is primitive, we have N H ( C ) = M . Thus, A ≤ M . Hence, J ( N ) is contained in M ≤ H . It follows that J ( H ) = J ( M ) in case (i), and J ( N ) = J ( M ∩ I ) in case (ii). The same argument with H replaced by K (and N replaced by L in case (ii)), however, also shows that J ( K ) = J ( M ) in case (i), and J ( L ) = J ( M ∩ I ) incase (ii). But then we have, in either case, that J ( N ) = J ( L ) is normal in both H and K - acontradiction. This completes the proof.We remark that the proof of Theorem 2.2 does not require the Thompson–Wielandt theorem.Also, part (i) above proves Theorem 1.2.We can now replace Steps (2) and (5) in the proof of the Sims conjecture with the followingeasy consequences of Theorem 2.2(i). We begin with Step (2): Corollary 2.3. Let G be an almost simple primitive permutation group, with point stabilizer H , and let h > be the length of a non-trivial h -orbit. Suppose that the socle X of G isa finite group of Lie type, and let p be the prime from the Thompson–Wielandt theorem. If | O p ( H ) ∩ Inndiag( X ) | is h -bounded, then | H | is h -bounded.Proof. Since Aut( X ) / Inndiag( X ) has a normal subgroup of order at most 6 with cyclic quotient,the Thompson–Wielandt theorem and the hypothesis of the corollary guarantee that H is cyclicby h -bounded. Theorem 2.2(i) then yields the result.Next, Step (5): Corollary 2.4. Let G be an almost simple primitive permutation group, with point stabilizer H , and let h > be the length of a non-trivial h -orbit. Let p be the prime from the Thompson–Wielandt theorem, and suppose that the socle X of G is a finite group of Lie type in p ′ charac-teristic, and that X has h -bounded (Lie) rank. Then | H | is h -bounded. roof. Write h = | H : H ∩ H g | , for an element g of G , and set K := H g , M := H ∩ K , and I := Inndiag( G ). Also, we may write X = O p ′ ( X σ ) for a simple algebraic group X of adjointtype and a Steinberg endomorphism σ of X . Since p is coprime to the defining characteristic of X , [9, II, Theorem 5.16] implies that O p ( H ) ∩ I normalizes of a maximal torus T in X . Then( O p ( H ) ∩ I ) / ( O p ( H ) ∩ I ∩ T ) is a subgroup of the Weyl group W of X . Since the Lie rank of X is h -bounded, the order of the group W is h -bounded. We deduce that (a) O p ( H ) ∩ I is abelianby h -bounded; (b) O p ( H ) ∩ I has h -bounded subgroup rank; and (c) O p ( H ) ∩ I has boundedderived length.It follows from (a) and the Thompson–Wielandt theorem that H ∩ I and K ∩ I = ( H ∩ I ) g are abelian by h -bounded. Since H > M < K is a primitive amalgam, Theorem 2.2(ii) thenimplies that H ∩ I and K ∩ I have h -bounded exponents. This, together with (b) and (c) above,implies that | O p ( H ) ∩ I | and | O p ( K ) ∩ I | are h -bounded. The result now follows from Corollary2.3.This completes our simplified proof of the Sims conjecture. References [1] P.J. Cameron, C.E. Praeger, J. Saxl, and G.M. Seitz. On the Sims conjecture and distancetransitive graphs. Bull. London Math. Soc. 15 (1983), 499–506.[2] P. S. Fan. Amalgams of finite groups and the Goldschmidt-Sims conjecture. Proceedingsof the Rutgers group theory year, 1983-1984 (ed. M. Aschbacher et al. CUP, Cambridge,1984), 161–166.[3] P. S. Fan. The Thompson-Wielandt theorem. Proc. Amer. Math. Soc. 97 (1986), 590–592.[4] P. S. Fan. Amalgams of prime index. J. Algebra. 98 (1986), 375–421.[5] D. Goldschmidt. Automorphisms of trivalent graphs. Ann. of Math. 111 (1980), 377–406.[6] W. Knapp. On the point stabilizer in a primitive permutation group. Math. Z. 133 (1973),137–168.[7] G. M. Seitz. The root subgroups for maximal tori in finite groups of Lie type. Pacific J.Math. 106 (1983), 153–244.[8] C.C. Sims. Graphs and finite permutation groups. Math. Z. 95 (1967), 76–86.[9] T. A. Springer and R. Steinberg. Conjugacy classes. Seminar on algebraic groups andrelated finite groups (eds A. Borel et al). Lecture Notes in Mathematics131