Bounding the degree of generic sharp transitivity

Tuna Altınel; Joshua Wiscons

doi:10.1515/jgth-2024-0152

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Bounding the degree of generic sharp transitivity

Tuna Altınel and Joshua Wiscons

Published/Copyright: March 4, 2025

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From the journal Journal of Group Theory Volume 28 Issue 6

Abstract

We show that a generically sharply 𝑡-transitive permutation group of finite Morley rank on a set of rank 𝑟 satisfies t ≤ r + 2 provided the pointwise stabilizer of a generic ( t − 1 ) -tuple is an 𝐿-group, which holds, for example, when this stabilizer is solvable or when r ≤ 5 . This makes progress towards establishing the natural bound on 𝑡 implied by the Borovik–Cherlin conjecture that every generically ( r + 2 ) -transitive permutation group of finite Morley rank on a set of rank 𝑟 is of the form PGL r + 1 ⁡ ( F ) acting naturally on P r ⁢ ( F ) . Our proof is assembled from three key ingredients that are independent of the main theorem – these address actions of Alt ⁡ ( n ) on 𝐿-groups of finite Morley rank, generically 2-transitive actions with abelian point stabilizers, and simple groups of rank 6.

1 Introduction

The study of generically 𝑡-transitive actions was popularized by Popov in the context of algebraic groups [26] and then by Borovik and Cherlin in the more general setting of groups of finite Morley rank [7], although the first substantial results appeared nearly two decades earlier with work of Gropp [22]. While ordinary 𝑡-transitivity requires that a group 𝐺 act transitively on X t off of the diagonal, generic 𝑡-transitivity only requires 𝐺 act transitively on X t off of some set of smaller dimension.

This is considerably more natural than ordinary 𝑡-transitivity. For example, the action of PGL n + 1 ⁡ ( F ) on P n ⁢ ( F ) is generically ( n + 2 ) -transitive (since the set of projective bases forms a single orbit), but it is not even 3-transitive once n > 1 . Generic 𝑡-transitivity also lends itself to a broader range of applications; this is particularly well demonstrated by its recent use in the theory of algebraic differential equations [20].

1.1 The problem

We work in the setting of groups of finite Morley rank, denoting Morley rank by rk ; the general theory can be found in [9]. As is common, our use of definable includes interpretable; it can simply be read as algebraic or constructible by those familiar with the algebraic setting but not with Morley rank. For a group acting on a set 𝑋, we say that ( G , X ) is a permutation group of finite Morley rank if the action is faithful and each of 𝐺, 𝑋, and the action of 𝐺 on 𝑋 are definable in some ambient structure of finite Morley rank.

Definition

A permutation group ( G , X ) of finite Morley rank is generically 𝑡-transitive if 𝐺 has an orbit 𝒪 on X t such that rk ⁡ ( X t ∖ O ) < rk ⁡ ( X t ) . The maximum such 𝑡 for which the action is generically 𝑡-transitive is called the degree of generic transitivity (or generic transitivity degree as in [26]), denoted gtd ⁡ ( G , X ) .

As we have already observed, generic 𝑡-transitivity exists very naturally for all 𝑡; however, one may still look to bound 𝑡 relative to a fixed 𝐺 or relative to actions on sets of a fixed rank.

In the former case, for fixed 𝐺, it is straightforward to see that gtd ⁡ ( G , X ) is bounded above by rk ⁡ G rk ⁡ X , but determining the precise value can be nontrivial. One outcome of Popov’s work was that, for a fixed connected nonabelian reductive group 𝐺, the maximum value of gtd ⁡ ( G , X ) , taken over all algebraic actions, is in fact associated to a natural action: the action of 𝐺 on the cosets of some maximal parabolic subgroup. Popov also computes gtd ⁡ ( G , X ) for all connected simple 𝐺 with 𝑋 the coset spaces of a maximal parabolic.

In the case of actions on sets of a fixed rank 𝑟, it is not immediately clear that there is any bound at all on the degree of generic transitivity, but, as shown by Borovik and Cherlin, there is. This is essentially the main result of [7] where they then raise the obvious question of what, precisely, the least upper bound is. In fact, they ask for more: to identify those actions that achieve the extreme.

Main Problem

Main Problem ([7, Problem 9])

Assume ( G , X ) is a transitive and generically 𝑡-transitive permutation group of finite Morley rank with 𝑋 of rank r ≥ 1 . Show that if t ≥ r + 2 , then ( G , X ) is isomorphic to PGL r + 1 ⁡ ( F ) acting naturally on P r ⁢ ( F ) (and hence t = r + 2 ).

This problem has only been solved in a handful of settings: when r ≤ 2 (by Hrushovski in rank 1 and the present authors in rank 2 [2]), in the theory of algebraically closed fields of characteristic 0 (by Freitag and Moosa [20, Theorem 6.3] leveraging Popov’s work), and in the theory of differentiably closed fields of characteristic 0 (by Freitag, Jimenez, and Moosa [19, Theorem 4.3]). It should be mentioned that the Main Problem has a natural analog for modules, and this has recently been solved in full generality by Berkman and Borovik [5].

The approach of the authors in rank 2 (and also of Berkman and Borovik for modules) indicates a canonical case division when attacking the Main Problem: is the action generically sharply𝑡-transitive or not.

Definition

A permutation group ( G , X ) of finite Morley rank is generically sharply 𝑡-transitive if it is generically 𝑡-transitive with large orbit 𝒪, and 𝐺 acts without fixing any tuple in 𝒪.

The point is that the natural action of PGL r + 1 ⁡ ( F ) on P r ⁢ ( F ) is in fact generically sharply ( r + 2 ) -transitive. As such, the “sharp version” of the Main Problem aims at identification, whereas the “non-sharp version” seeks a contradiction. The sharp version itself can be broken down further: establish the desired bound on 𝑡 and then proceed with identification. This naturally breaks the Main Problem into three sub-problems.

Problem 1

Problem 1 (Sharpness)

Assume ( G , X ) is a generically 𝑡-transitive permutation group of finite Morley rank with rk ⁡ X = r . Show that if t ≥ r + 2 , then the action must be generically sharply𝑡-transitive.

Problem 2

Problem 2 (Bound)

Assume ( G , X ) is a generically sharply 𝑡-transitive permutation group of finite Morley rank with rk ⁡ X = r . Show that if t ≥ r + 2 , then t = r + 2 .

Problem 3

Problem 3 (Identification)

Assume ( G , X ) is a transitive and generically sharply 𝑡-transitive permutation group of finite Morley rank with rk ⁡ X = r . Show that if t = r + 2 , then ( G , X ) ≅ ( PGL r + 1 ⁡ ( F ) , P r ⁢ ( F ) ) .

1.2 The theorem

At present, little has been done around Problem 1, but its solution for modules provides a sliver of hope. Earlier work of the authors addressed aspects of Problem 3 in arbitrary rank [3]; here, we study Problem 2, which previously was only handled when r ≤ 2 as a consequence of [2] (see also the early work of Gropp [22]).

Our main result is rank free but comes with a condition on the stabilizer of a generic ( t − 1 ) -tuple from 𝑋. When ( G , X ) ≅ ( PGL r + 1 ⁡ ( F ) , P r ⁢ ( F ) ) , this stabilizer is precisely a maximal torus of 𝐺, and for our result, we “simply” ask that the stabilizer does not involve a certain (conjecturally empty) class of simple groups. In particular, the condition will automatically be met if this stabilizer is known to be solvable. The precise framework we work in is that of 𝐿-groups.

Definition

Let 𝐻 be a group of finite Morley rank.

𝐻 has degenerate type if it has no involutions;
𝐻 has even type if it contains an infinite elementary abelian 2-group but no Prüfer 2-group;
𝐻 has odd type if it contains a Prüfer 2-group but no infinite elementary abelian 2-group;
𝐻 has mixed type if it contains an infinite elementary abelian 2-group and a Prüfer 2-group.

We say 𝐻 is an 𝐿-group if every definable simple section of 𝐻 that is of odd type is isomorphic to an algebraic group over an algebraically closed field.

It is known – and nontrivial – that every connected group of finite Morley rank has one of the four types listed above [10, 6] and that the infinite simple ones cannot have mixed type [1].

Remark

The Algebraicity Conjecture of Cherlin and Zilber posits that every infinite simple group of finite Morley rank is isomorphic to an algebraic group over an algebraically closed field. This analog of the classification of the finite simple groups remains far from being established, but remarkably, it has been confirmed in even and mixed type (with no simple groups of the latter type) [1]. Attacking degenerate-type simple groups amounts to proving a Feit–Thompson result in this context, which has not seen much progress. The 𝐿-hypothesis aims to push the theory forward without assuming knowledge of degenerate-type groups. In the setting of Problem 2, this is made possible by [4].

Here is our main result.

Theorem

Assume ( G , X ) is a generically sharply 𝑡-transitive permutation group of finite Morley rank with rk ⁡ X = r . Let 𝐻 be the pointwise stabilizer of a generic ( t − 1 ) -tuple. If 𝐻 is an 𝐿-group, then t ≤ r + 2 .

We stress that the 𝐿-hypothesis is not placed on 𝐺 but rather only on 𝐻, which conjecturally should be a torus when t ≥ r + 2 . As mentioned above, it is [4] that provides traction for the 𝐿-hypothesis; in particular, [4, Corollary] tells us at the outset that either 𝐻 is solvable or contains involutions.

Remark

Our work also yields information when t = r + 2 . In this extremal case, Propositions 1 and 5 can be used to show that a generic ( t − 1 ) -tuple must necessarily be solvable except possibly when r = 4 . Analysis similar to what we undertake for r = 3 in Proposition 5 (leveraging the significant Proposition 3) may establish solvability when r = 4 , but we do not take it up here.

Using the classification of groups of rank at most 5 [27, 12, 31, 16, 18] and the classification of the simple groups of finite Morley rank of even type [1], the Theorem immediately yields the following.

Corollary

Assume ( G , X ) is a generically sharply 𝑡-transitive permutation group of finite Morley rank with rk ⁡ X = r . Then t ≤ r + 2 provided any one of the following hold:

r ≤ 5 ;
𝐺 has even type;
the stabilizer of a generic ( t − 1 ) -tuple is solvable.

1.3 The strategy

Let us outline our approach to the Theorem. Assume we are in the setting of the Theorem with ( 1 , … , t ) a generic 𝑡-tuple in X t and G [ t − 1 ] the pointwise stabilizer of 1 , … , t − 1 . Let Σ be the setwise stabilizer of 1 , … , t and Σ t ≤ Σ the stabilizer of 𝑡 in Σ. Generic sharp 𝑡-transitivity readily implies Σ ≅ Sym ⁡ ( t ) and G [ t − 1 ] is connected of rank 𝑟. By [2, Lemma 4.27], Σ t acts faithfully on G [ t − 1 ] . Towards a contradiction, we assume that t ≥ r + 3 . The main point then is that G [ t − 1 ] carries a faithful action of the relatively large symmetric group Sym ⁡ ( r + 2 ) , and this is restrictive.

In Section 2.1, we study actions of Alt ⁡ ( n ) and Sym ⁡ ( n ) on connected nonsolvable𝐿-groups of rank 𝑟 and show in Proposition 1 that typically n ≤ r . When t ≥ r + 3 , this yields that either (1) G [ t − 1 ] is solvable, or (2) we are in the exceptional case of Σ t ≅ Sym ⁡ ( 5 ) and G [ t − 1 ] ≅ PGL 2 ⁡ ( F ) with char ⁡ F = 5 (which is an expected complication as Sym ⁡ ( 5 ) ≅ PGL 2 ⁡ ( F 5 ) ). To push further, both cases require us to work with a larger chunk of 𝐺 than just G [ t − 1 ] , but it turns out that G [ t − 2 ] (the pointwise stabilizer of 1 , … , t − 2 ) suffices.

When G [ t − 1 ] is solvable, the condition t ≥ r + 3 forces G [ t − 1 ] to be an elementary abelian 𝑝-group. This configuration is studied at the level of G [ t − 2 ] and eliminated by Proposition 2, which looks at generically 2-transitive actions with abelian point stabilizers. The case of G [ t − 1 ] ≅ PGL 2 ⁡ ( F ) corresponds to rk ⁡ G [ t − 2 ] = 6 , and to analyze this, we undertake a rather general study of simple groups of rank 6 in Section 2.3. This culminates with Proposition 3, showing that such groups are so-called N ∘ ∘ -groups in the sense of [15], which we leverage to obtain a contradiction.

We emphasize that the key ingredients of our proof – Propositions 1, 2, and 3 – live beyond the specific context of the Theorem and should be interesting in their own right.

1.4 Regarding the 𝐿-hypothesis

We would like to highlight that the 𝐿-hypothesis in the Theorem is only used when we are applying Proposition 1, or more to the point, removing the hypothesis from Proposition 1 would remove it from the Theorem.

Despite some effort trying to free Proposition 1 from the 𝐿-hypothesis, we were not able to realize it. What is needed is an extension of Corollary 2.6 from actions of Alt ⁡ ( n ) on simple algebraic groups to actions on simple odd-type groups of finite Morley rank. One angle we entertained was to adapt the analysis of actions on degenerate-type groups from [4]; there, the approach is inductive, driven by the fact that a degenerate-type group acted upon by a Klein 4-group 𝐾 is generated by the centralizers of the involutions in 𝐾 (see [6, Theorem 5]). Our attempt to generalize this to actions on odd-type groups failed to find traction, and this may well be a difficult problem.

Although the value is unclear, one could try replacing the 𝐿-hypothesis of Proposition 1 with the assumption that the “four-group generation” result holds for the simple definable sections of 𝐻, leading to an analysis of 𝑉-groups (as 𝐾 is already taken). We thought very briefly on this. A 𝑉-assumption moves the analysis of [4] along quite smoothly until it requires a lower bound on the corank of a proper definable subgroup (see [4, Corollary 2.5]), which uses the fact that the degree of generic transitivity of any action of a connected degenerate-type group is at most 1. For odd-type groups, there is no such universal bound, but something can still be said. For example, since the degree of generic transitivity of a connected solvable group is at most 2, one obtains that the degree of generic transitivity of a connected minimal simple group is at most 3. This may be enough to prove a statement along the following lines: if Alt ⁡ ( n ) , with 𝑛 sufficiently large, acts on a connected minimal simple 𝑉-group 𝐻 of finite Morley rank, then n ≤ rk ⁡ H + 2 .

There is also the question of trying to extend Corollary 2.6 to simple L ∗ -groups 𝐺 of odd type. The L ∗ -hypothesis assumes all proper definable sections of 𝐺 are 𝐿-groups but makes no claim on 𝐺 itself. The case of simple L ∗ -groups with sufficiently high Prüfer rank has recently been reviewed and clarified by Cherlin: assuming such a 𝐺 has Prüfer 2-rank at least 3 and that there are certain internal restrictions on how 2-tori act (the NTA₂ hypothesis), then either 𝐺 is algebraic or contains a strongly embedded subgroup [13]. As such, a test case for extending Corollary 2.6 to the L ∗ -setting might be when 𝐺 has high Prüfer 2-rank and a strongly embedded subgroup. Or, instead focusing on low Prüfer 2-rank, one could try to analyze actions on groups of type CiBo₁, CiBo₂, or CiBo₃ (as defined in Section 2.3) with an initial focus on the minimal simple case.

2 Three key ingredients

Here we develop the main ingredients of our proof of the Theorem, as laid out in Section 1.3. These results are independent of the Theorem and address actions of Alt ⁡ ( n ) and Sym ⁡ ( n ) on 𝐿-groups of finite Morley rank (Section 2.1), generically 2-transitive actions with abelian point stabilizers (Section 2.2), and simple groups of rank 6 (Section 2.3).

2.1 Actions of Alt ⁡ ( n ) on nonsolvable groups

Our work here focuses on the following conjecture, which previously had only been addressed for actions on groups of degenerate type [4].

Conjecture

Conjecture (see [14, Section 1.2])

Suppose that Alt ⁡ ( n ) acts definably and faithfully by automorphisms on a nonsolvable connected group 𝐻 of finite Morley rank. Then n ≤ rk ⁡ H for sufficiently large 𝑛.

The solvable version of the conjecture allows for n ≤ rk ⁡ H + 2 and was addressed in [4, Lemma 2.7] with the critical case of abelian 𝐻 taken care of by the analysis of [14].

Leveraging the algebraic theory, we solve the conjecture for actions on 𝐿-groups; this is Proposition 1. Our proof consists to a large degree of simply collecting and organizing known results. As indicated in Corollary 2.5, the bound of rk ⁡ H on 𝑛 is typically rather weak and should be more on the order of rk ⁡ H for generic 𝑛.

Proposition 1

Assume Alt ⁡ ( n ) or Sym ⁡ ( n ) acts definably and faithfully by automorphisms on a nonsolvable connected 𝐿-group 𝐻 of finite Morley rank. If n ≥ 9 , then n ≤ rk ⁡ H . If the acting group is Sym ⁡ ( n ) , then n ≥ 5 also implies n ≤ rk ⁡ H unless n = 5 and H ≅ Z ⁢ ( H ) × PGL 2 ⁡ ( F ) with char ⁡ F = 5 .

Our approach will consider projective representations of Alt ⁡ ( n ) of minimal dimension for which we first collect results on the linear representations of Alt ⁡ ( n ) for n ≥ 8 and its unique Schur cover, which we denote by Alt ̂ ⁢ ( n ) .

Fact 2.1

Fact 2.1 ([28, 29])

Let n ≥ 9 . Suppose Alt ⁡ ( n ) ≤ GL d ⁡ ( F ) for 𝐹 an algebraically closed field. Then d ≥ n − 1 − κ n , where κ n = 1 if char ⁡ F divides 𝑛 and κ n = 0 otherwise. Moreover, the same bound holds for n = 7 , 8 provided char ⁡ F ≠ 2 .

Fact 2.2

Fact 2.2 ([25, Theorem A])

Let n ≥ 8 . Suppose Alt ̂ ⁢ ( n ) ≤ GL d ⁡ ( F ) for 𝐹 an algebraically closed field of characteristic not 2. Then d ≥ 2 ⌊ ( n − 2 − κ n ) / 2 ⌋ , where κ n = 1 if char ⁡ F divides 𝑛 and κ n = 0 otherwise.

The previous facts combine to yield the following.

Corollary 2.3

Let n ≥ 9 . Suppose Alt ⁡ ( n ) ≤ PGL d ⁡ ( F ) for 𝐹 an algebraically closed field. Then n ≤ d + 2 .

We now consider embeddings of Alt ⁡ ( n ) into other simple algebraic groups. Our approach is cheap: given a simple algebraic group 𝐺, we embed 𝐺 into PGL d ⁡ ( F ) and then apply Corollary 2.3.

Fact 2.4

Suppose 𝐺 is a (group-theoretically) simple algebraic group over an algebraically closed field 𝐹. Then Table 1 gives the algebraic dimension of 𝐺 and minimal 𝑑 such that 𝐺 embeds into PGL d ⁡ ( F ) .

Table 1

Dimensions of the simple algebraic groups and minimal 𝑑 such that they embed into PGL d ⁡ ( F )

Type of 𝐺	dim ( G )	𝑑 such that 𝐺 embeds in PGL d ⁡ ( F )
A ⁢ ( ℓ )	( ℓ + 1 ) 2 − 1	ℓ + 1
B ⁢ ( ℓ )	2 ⁢ ℓ 2 + ℓ	2 ⁢ ℓ + 1
C ⁢ ( ℓ )	2 ⁢ ℓ 2 + ℓ	2 ⁢ ℓ
D ⁢ ( ℓ )	2 ⁢ ℓ 2 − ℓ	2 ⁢ ℓ
E ⁢ ( 6 )	78	27
E ⁢ ( 7 )	133	56
E ⁢ ( 8 )	248	248
F ⁢ ( 4 )	52	26
G ⁢ ( 2 )	14	7

A few remarks are in order to justify the data contained in Table 1. To obtain the second column, it suffices to compute the cardinality of the set of positive roots in the corresponding linear algebraic group. The dimension of the group is then twice this cardinality plus the dimension of a maximal torus. The numerical data for this can be found in [23, Section 12.2] (see also [24, Section 28]). A compact exposition of the dimensions in the third column, in a more general setting, can be found in [11] with the justification given in [11, Theorem 1.2]. (It seems that a slight typo has found its way into the formula for F 4 on [11, page 672], but the argument on [11, page 676] rectifies it.)

Table 2

Upper bounds on 𝑛 when Alt ⁡ ( n ) embeds into 𝑟-dimensional classical groups

Type of 𝐺	f ⁢ ( r )
A ⁢ ( ℓ )	2 + r + 1
B ⁢ ( ℓ )	5 + 8 ⁢ r + 1 2
C ⁢ ( ℓ )	3 + 8 ⁢ r + 1 2
D ⁢ ( ℓ )	5 + 8 ⁢ r + 1 2

Corollary 2.5

Let n ≥ 9 . Suppose Alt ⁡ ( n ) ≤ G for 𝐺 an 𝑟-dimensional, simple algebraic group over an algebraically closed field. Then n ≤ r , and if 𝐺 is a classical group, then n ≤ f ⁢ ( r ) , where f ⁢ ( r ) is given in Table 2.

Proof

Corollary 2.3 and Fact 2.4 resolve this for all cases except E ⁢ ( 8 ) , so suppose 𝐺 is of type E ⁢ ( 8 ) . The Weyl group involves only the primes up to 7, so if E < G is an elementary abelian 𝑝-group with p > 7 and p ≠ char ⁡ F , then 𝐸 is toral, implying that the 𝑝-rank of such an 𝐸 is at most 8.

If n ≥ 99 , then Alt ⁡ ( n ) contains an elementary abelian 11-group of 11-rank 9, showing that n < 99 when char ⁡ F ≠ 11 . And if char ⁡ F = 11 , we may instead consider elementary abelian 13-groups to see that n < 117 .

The proof is complete, but we note that this method can be applied to the primes 2 and 3 (to produce much smaller bounds) provided one also accounts for nontoral such 𝐸, for which [21] gives the necessary information. ∎

Corollary 2.6

Assume Alt ⁡ ( n ) or Sym ⁡ ( n ) acts definably and faithfully on a simple algebraic group 𝐺 over an algebraically closed field. If n ≥ 9 , then n ≤ dim G . If the acting group is Sym ⁡ ( n ) , then n ≥ 5 also implies n ≤ dim G unless n = 5 and G = PGL 2 ⁡ ( F ) with char ⁡ F = 5 .

Proof

By [1, Chapter II, Fact 2.25] (generalizing [24, Theorem 27.4]), the acting group is contained in Inn ⁡ ( G ) ⋅ Γ for Γ the group of graph automorphisms relative to a fixed maximal torus and Borel subgroup, so Alt ⁡ ( n ) embeds into 𝐺 in the cases under consideration. By Corollary 2.5, it only remains to address when 5 ≤ n ≤ 8 and the acting group is Sym ⁡ ( n ) . Let r = dim G . If r ≥ 8 , there is nothing to show, so we may assume 𝐺 is of type A ⁢ ( 1 ) . In particular, r = 3 , and G = PGL 2 ⁡ ( F ) has no graph automorphisms. Thus Sym ⁡ ( n ) embeds into PGL 2 ⁡ ( F ) . This is now quite classical. Via the adjoint representation of PGL 2 ⁡ ( F ) , we have a faithful 3-dimensional representation of Sym ⁡ ( n ) , so the only possibility is n = 5 = char ⁡ F (see for example [17]). ∎

Remark

Since the Morley rank of an algebraic group 𝐺 is always greater than or equal to its algebraic dimension, the bound n ≤ dim G appearing in Corollary 2.6 can be replaced by n ≤ rk ⁡ G .

Proof of Proposition 1

We will denote by 𝑋 the acting group, 𝑛 the degree of the permutation group 𝑋, and 𝑟 the rank of 𝐻. Also, set X ′ = [ X , X ] = Alt ⁡ ( n ) , and let σ ⁢ ( H ) denote the solvable radical of 𝐻.

We begin by addressing the exceptional case of n = 5 .

Claim 1

If X = Sym ⁡ ( 5 ) and r < 5 , then H ≅ Z ⁢ ( H ) × PGL 2 ⁡ ( F ) with 𝐹 of characteristic 5.

Proof of claim

Since 𝐻 is nonsolvable and connected groups of Morley rank at most 2 are solvable [12], 𝑟 is either 3 or 4.

First assume r = 3 . Then 𝐻 is quasisimple with a finite center. Notice that 𝑋 acts faithfully on H / Z ⁢ ( H ) ; otherwise, the connected group [ X ′ , H ] is contained in Z ⁢ ( H ) and hence is trivial (since Z ⁢ ( H ) is finite), contradicting faithfulness of the action on 𝐻. Now, simple groups of rank 3 are known to be algebraic [12, 18], so Corollary 2.6 applies to H / Z ⁢ ( H ) . Thus H / Z ⁢ ( H ) ≅ PGL 2 ⁡ ( F ) with char ⁡ F = 5 . Moreover, by the theory of central extensions (see [1, Chapter II, Lemma 2.21]), 𝐻 is algebraic, hence of the form PGL 2 ⁡ ( F ) or SL 2 ⁡ ( F ) . In either case, 𝑋 must act by inner automorphisms (see [1, Chapter II, Lemma 2.27]), but SL 2 ⁡ ( F ) does not embed Sym ⁡ ( 5 ) . Thus Z ⁢ ( H ) = 1 , and 𝐻 has the desired structure.

Next consider when r = 4 . By [4], 𝐻 contains involutions, and its structure is given by [31]: H = Z ∘ ⁢ ( H ) ∗ Q with 𝑄 quasisimple of rank 3. Since Z ∘ ⁢ ( H ) has rank 1, it is centralized by X ′ , so 𝑋 must act faithfully on 𝑄. Our analysis above then shows Q ≅ PGL 2 ⁡ ( F ) in characteristic 5, which in turn forces H = Z ⁢ ( H ) × Q . ∎

For the remainder of the proof, we assume n ≥ 6 and proceed by contradiction. Thus we assume r < n , and we choose 𝐻 to be a counterexample to Proposition 1 of minimal rank 𝑟.

Claim 2

σ ∘ ⁢ ( H ) = 1 .

Proof of claim

Assume σ ∘ ⁢ ( H ) ≠ 1 . Then we have rk ⁡ ( H / σ ∘ ⁢ ( H ) ) < r , and since 𝑋 acts on H / σ ∘ ⁢ ( H ) , minimality of 𝑟 implies that this action is not faithful. Thus [ X ′ , H ] ≤ σ ∘ ⁢ ( H ) . Now, if we also have [ X ′ , σ ∘ ⁢ ( H ) ] = 1 , then [ X ′ , [ X ′ , H ] ] = 1 , and the three subgroups lemma implies that the group [ X ′ , X ′ ] = X ′ centralizes 𝐻, against faithfulness of the action on 𝐻. Thus [ X ′ , σ ∘ ⁢ ( H ) ] ≠ 1 , so 𝑋 acts faithfully on σ ∘ ⁢ ( H ) .

Then [4, Lemma 2.7] implies rk ⁡ ( σ ∘ ⁢ ( H ) ) ≥ n − 2 in all cases except possibly when X = Alt ⁡ ( 9 ) . However, if X = Alt ⁡ ( 9 ) and rk ⁡ ( σ ∘ ⁢ ( H ) ) < 7 , we can use that the minimal rank of a faithful connected Alt ⁡ ( 9 ) -module is 7 (see [14, main result and the final remark of Section 3.3]) to eliminate this case. Thus

rk ⁡ ( σ ∘ ⁢ ( H ) ) ≥ n − 2 ,

meaning that rk ⁡ ( H / σ ∘ ⁢ ( H ) ) ≤ 2 . Since all connected groups of Morley rank at most 2 are solvable, this is a contradiction, so σ ∘ ⁢ ( H ) = 1 . ∎

Claim 3

We may assume 𝐻 is semisimple: there exist k ≥ 1 and connected simple groups of finite Morley rank H 1 , … , H k such that H = H 1 × ⋯ × H k .

Proof of claim

Since σ ∘ ⁢ ( H ) = 1 , we find (as in the proof of Claim 1) that 𝑋 acts faithfully on H / σ ⁢ ( H ) . Thus H / σ ⁢ ( H ) is also a counterexample of rank 𝑟, and the socle of 𝐻 is as well. Thus we may assume 𝐻 equals its socle, which is semisimple by [9, Theorem 7.8]. ∎

Claim 4

If n ≥ 9 , we have a contradiction.

Proof of claim

Since 𝐻 is an 𝐿-group, each H i is either a simple linear algebraic group over an algebraically closed field or of degenerate type. By the inductive assumption, n > r ≥ k (in fact, n > r ≥ 3 ⁢ k since the H i are not solvable). As X ′ is simple and permutes the set { 1 , … , k } , the only possible action is the trivial one. Thus X ′ normalizes each H i . However, rk ⁡ H i ≤ r < n , so X ′ centralizes each H i using Corollary 2.6 in the algebraic case and [4] in the degenerate case. This contradicts faithfulness. ∎

Claim 5

If 6 ≤ n ≤ 8 and X = Sym ⁡ ( n ) , we have a contradiction.

Proof of claim

By the contradictory assumption, r ≤ 7 . Since connected groups of Morley rank at most 2 are solvable, 𝑘 is at most 2.

We first eliminate the case when k = 2 . If k = 2 then one of the groups, say H 1 , is of rank 3 and the other is of rank 4. As a result, 𝑋 cannot permute these two groups so normalizes each one. Minimality of 𝑟 implies that the action of 𝑋 on both components has a nontrivial kernel, which must be X ′ . It follows that the action of 𝑋 on 𝐻 is not faithful, a contradiction.

Thus k = 1 and 𝐻 is simple. As in Claim 4, Corollary 2.6 and [4] provide a contradiction. ∎

The proof of Proposition 1 is now complete. ∎

2.2 Actions with virtually abelian point-stabilizers

Proposition 2

Let ( G , X ) be a transitive and generically 2-transitive permutation group of finite Morley rank with 𝐺 connected. If G x ∘ is abelian, then every virtually definably primitive quotient X ̄ , with corresponding kernel 𝑁, satisfies ( G / N , X ̄ ) ≅ ( AGL 1 ⁡ ( L ) , L + ) and G x ⁢ N / N ≅ L × for 𝐿 an algebraically closed field.

Setup

Throughout this section, we assume that ( G , X ) is a transitive permutation group of finite Morley rank.

Lemma 2.7

Let ( G , X ) be generically 𝑡-transitive with t ≥ 2 . Assume ( G 1 , … , t − 1 ) ∘ is abelian for ( 1 , … , t ) in the generic orbit of 𝐺 on X t . Then every definable quotient X ̄ , with corresponding kernel 𝑁, satisfies rk ⁡ ( G 1 , … , t − 1 ∩ N ) = rk ⁡ X − rk ⁡ X ̄ . In particular, rk ⁡ X > rk ⁡ X ̄ implies rk ⁡ N > rk ⁡ X − rk ⁡ X ̄ .

Proof

Let X ̄ and 𝑁 be as given; then 𝑋 and X ̄ are connected by [7, Lemma 1.8]. Set H : = ( G 1 , … , t − 1 ) ∘ . Since 𝐻 is abelian and generically transitive on 𝑋, 𝐻 is generically regular (using [7, Lemma 1.6]), so rk ⁡ H = rk ⁡ X . By [7, Lemma 6.1], the same is true on X ̄ modulo the kernel, so rk ⁡ H ⁢ N / N = rk ⁡ X ̄ . Thus

rk ⁡ H − rk ⁡ H ∩ N = rk ⁡ X ̄ ,

and all together, we have rk ⁡ H ∩ N = rk ⁡ H − rk ⁡ X ̄ = rk ⁡ X − rk ⁡ X ̄ .

For the final point, if rk ⁡ X > rk ⁡ X ̄ , then our work shows N ∘ ≠ 1 , so faithfulness implies that N ∘ ⊈ H . Thus rk ⁡ N > rk ⁡ H ∩ N . ∎

Lemma 2.8

Let ( G , X ) be generically 2-transitive. If G x is abelian-by-finite, then every definable quotient X ̄ , with corresponding kernel 𝑁, satisfies G x ̄ ∘ = G x ∘ ⁢ N , so G x ̄ / N is also abelian-by-finite.

Proof

Let X ̄ and 𝑁 be as given. We proceed by induction on the difference d = rk ⁡ X − rk ⁡ X ̄ . If d = 0 , the index of G x in G x ̄ is finite, and we are done.

Assume d ≥ 1 . Then, by Lemma 2.7, N ∘ is nontrivial, and as N ∘ is not contained in rk ⁡ G x (by faithfulness), rk ⁡ G x ⁢ N > rk ⁡ G x . Now let X ̃ be the quotient of 𝑋 determined by G x ⁢ N and 𝐾 the corresponding kernel. We simultaneously study the actions of 𝐺 on 𝑋, X ̃ , and X ̄ , which correspond to the sequence of stabilizers G x < G x ̃ ≤ G x ̄ . We show that the hypotheses on 𝑋 pass to X ̃ , to which induction applies to obtain the result for X ̄ .

Since N ≤ G x ⁢ N = G x ̃ , normality of 𝑁 forces N ≤ K , and as K ≤ G x ̃ ≤ G x ̄ , we also have K ≤ N , hence equality. Thus G x ̃ / K = G x ̃ / N = G x ⁢ N / N is abelian-by-finite, so ( G / K , X ̃ ) satisfies the hypotheses of the lemma. Induction applies, and as X ̄ is a quotient of X ̃ , G x ̄ ∘ / K = G x ̃ ∘ ⁢ N / K . Using the fact that K = N and G x ̃ = G x ⁢ N , we obtain the desired result. ∎

Proof of Proposition 2

Taking X ̄ and 𝑁 as stated, Lemma 2.8 shows that the point stabilizers of ( G / N , X ̄ ) are again abelian-by-finite, so by [31, Lemma 3.7], they are in fact connected and abelian. Then [31, Proposition 3.8] applies to give the desired structure of ( G / N , X ̄ ) and hence the structure of G x ⁢ N / N as well since G x ̄ = G x ∘ ⁢ N (from Lemma 2.8). ∎

Though we do not make use of it here, we record a corollary of Proposition 2 that appears quite relevant to continued work on the Main Problem.

Corollary 2.9

Let ( G , X ) be generically 2-transitive with 𝐺 connected. If G x ∘ is a good torus, then 𝐺 has a definable subnormal series

G = N 0 ⁢ ⊵ ⁢ N 1 ⁢ ⊵ ⁢ ⋯ ⁢ ⊵ ⁢ N m = 1

such that N i / N i + 1 ≅ AGL 1 ⁡ ( L i ) with each L i an algebraically closed field.

Proof

Fix x , y in general position.

We begin with a few observations. First, [31, Lemma 3.7] implies G x is connected. It also implies G x is a maximal good torus, which we now show. Indeed, if not, G x and G y would be properly contained in good tori T x and T y , which must intersect nontrivially by rank considerations. Further, T x and T y would generate 𝐺 since G x and G y do (see for example [2, Lemma 4.12]), and this would force 𝐺 to have a nontrivial center, against [31, Lemma 3.7]. For the same reason, G x ∩ G y = 1 , so the action is generically sharply2-transitive.

We apply Proposition 2 to the action of 𝐺 on 𝑋. Letting 𝑁 be the kernel of a virtually definably primitive quotient X ̄ , we have G / N ≅ AGL 1 ⁡ ( L ) . Since Proposition 2 applies to any virtually definably primitive quotient, we may assume 𝑁 is connected (by passing from the quotient determined by G x ⁢ N to the one determined by G x ⁢ N ∘ ).

Now let 𝒪 be the orbit of 𝑁 on 𝑋 that contains 𝑥. We show that Proposition 2 applies to the action of 𝑁 on 𝒪 and then conclude by induction. Only faithfulness and generic 2-transitivity need to be verified.

Now, N x and N y must be maximal good tori of 𝑁, so by conjugacy of maximal tori in 𝑁, N y = N y ′ for some y ′ ∈ O . This was the key point. Since G x ∩ G y = 1 , we find that N x ∩ N y ′ = 1 , implying, in particular, that the action of 𝑁 on 𝒪 is faithful. This also can be used to show that the action of 𝑁 on 𝒪 is generically 2-transitive. Let r = rk ⁡ X and s = rk ⁡ X ̄ . We have that

r − s = rk ⁡ G x ̄ / G x = rk ⁡ G x ⁢ N / G x = rk ⁡ N / N x = rk ⁡ O ;

similarly, we find that rk ⁡ N x = r − s = rk ⁡ O . Thus, since N x ∩ N y ′ = 1 , the orbit of N x containing y ′ is generic, establishing generic 2-transitivity of ( N , O ) . ∎

2.3 Groups of rank 6

Our analysis in the proof of the Theorem leads to an exceptional case for which we need rather detailed information about quasisimple groups of Morley rank 6. We prove a reasonably strong result for simple groups of Morley rank 6 that places the quasisimple groups within the N ∘ ∘ -framework of Deloro and Jaligot.

Definition

A group 𝐺 of finite Morley rank is called an N ∘ ∘ -group if N G ∘ ⁢ ( A ) remains solvable whenever 𝐴 is an infinite, definable, connected, and solvable subgroup of 𝐺.

Proposition 3

If 𝐺 is a simple group of Morley rank 6, then 𝐺 has no rank 5 subgroups and every connected rank 4 subgroup is solvable. Consequently, every quasisimple group of Morley rank 6 with finite center is an N ∘ ∘ -group.

In reading Proposition 3, bear in mind that, conjecturally, the only simple groups of Morley rank 6 are those of the form PSL 2 ⁡ ( L ) with 𝐿 of rank 2. Also, we note that our use of the proposition occurs in a restricted setting where we will be able to show that connected centralizers of involutions are solvable; this will allow us to bring to bear the main result of [15].

Fact 2.10

Fact 2.10 ([15])

Let 𝐺 be an infinite connected nonsolvable N ∘ ∘ -group of finite Morley rank of odd type. Further, assume that C ∘ ⁢ ( i ) is solvable for all i ∈ I ⁢ ( G ) . Then one of the following holds.

CiBo: C ∘ ⁢ ( i ) is a Borel subgroup of 𝐺, and either
1. m 2 ⁡ ( G ) = 1 , C ⁢ ( i ) is a self-normalizing Borel subgroup of 𝐺;
2. m 2 ⁡ ( G ) = 2 , pr 2 ⁡ ( G ) = 1 , C ∘ ⁢ ( i ) is an abelian Borel subgroup of 𝐺 inverted by any ω ∈ C ⁢ ( i ) ∖ { i } , and rk ⁡ G = 3 ⋅ rk ⁡ C ∘ ⁢ ( i ) ; or
3. m 2 ⁡ ( G ) = pr 2 ⁡ ( G ) = 2 , C ⁢ ( i ) is a self-normalizing Borel subgroup of 𝐺.
Algebraic: G ≅ PSL 2 ⁡ ( K ) .

Proof of Proposition 3

Assume 𝐺 is a simple group of Morley rank 6. That 𝐺 has no rank 5 subgroups is due to Hrushovski; see [9, Theorem 11.98]. So we aim to show that connected rank 4 subgroups are solvable. Once done, the final sentence of Proposition 3 then follows from the fact that connected groups of rank at most 2 are solvable.

Now suppose M < G is nonsolvable and connected of rank 4. Our general strategy is as follows: (1) use the presence of the nonsolvable 𝑀 of rank 4 to produce a well-structured solvable 𝐷 of rank 4, (2) study ( G , G / D ) to identify 𝐺 as PSL 2 ⁡ ( F ) , and (3) note that PSL 2 ⁡ ( F ) has no subgroup like 𝑀.

Claim 1

If H < G is connected (solvable or not) of rank 4, then ( G , G / H ) is generically 3-transitive and 𝐻 is nonnilpotent.

Proof of claim

Suppose H < G with rk ⁡ H = 4 . Then 𝐻 is a maximal connected subgroup, so [7, Proposition 2.3] implies

s ⋅ gtd ⁡ ( G , G / H ) ≤ rk ⁡ G ≤ s ⋅ gtd ⁡ ( G , G / H ) + s ⁢ ( s − 1 ) / 2

for s = rk ⁡ G / H . Thus we have 2 ⋅ gtd ⁡ ( G , G / H ) ≤ 6 ≤ 2 ⋅ gtd ⁡ ( G , G / H ) + 1 , implying that gtd ⁡ ( G , G / H ) = 3 . Consequently, gtd ⁡ ( H , G / H ) = 2 , so [2, Proposition 4.24] shows 𝐻 is nonnilpotent. ∎

The claim implies that ( M , G / M ) is generically 2-transitive, so our chosen 𝑀 contains involutions. Its structure is then given by [31, Corollary A] together with [18]: M = Q ∗ Z with Q ≅ ( P ) ⁢ SL 2 ⁡ ( K ) and Z = Z ∘ ⁢ ( M ) (and rk ⁡ Z = 1 ).

Claim 2

Let M g be a generic conjugate of 𝑀. Set

A : = ( M ∩ M g ) ∘ and C : = C G ∘ ( A ) .

Then

rk ⁡ A = rk ⁡ C = 2 and rk ⁡ ( A ∩ C ) = 0 ;
C = ⟨ Z , Z g ⟩ ;
A / Z ⁢ ( A ) ≅ AGL 1 ⁡ ( K ) and C / Z ⁢ ( C ) ≅ AGL 1 ⁡ ( L ) for fields 𝐾 and 𝐿;
Z ≅ L × ≅ Z g .

Proof of claim

We first claim that neither 𝑍 nor Z g are contained in 𝐴. To see this, first note that 𝐴 is the connected component of a generic 2-point stabilizer in the action of 𝐺 on G / M . Thus if Z ≤ A , then 𝑍 fixes a point in the generic orbit of 𝑀 on G / M , so as 𝑍 is central in 𝑀, this implies 𝑍 fixes a generic subset of G / M , hence is in the kernel of the action, a contradiction to the simplicity of 𝐺. We conclude that Z ≰ A and similarly that Z g ≰ A .

We next show that A / Z ⁢ ( A ) ≅ AGL 1 ⁡ ( K ) . First, observe that rk ⁡ A = 2 ; this follows from the fact that 𝐴 is the connected component of a generic 2-point stabilizer in ( G , G / M ) , which is generically 3-transitive. Now, since rk ⁡ Z = 1 and Z ≰ A , we have that rk ⁡ ( Z ∩ A ) = 0 , so the image of 𝐴 in M / Z has rank 2. By the structure of 𝑀, A / Z ⁢ ( A ) must be isomorphic to a Borel of PSL 2 ⁡ ( K ) , so A / Z ⁢ ( A ) ≅ AGL 1 ⁡ ( K ) , as desired.

We now look at 𝐶. By the structure of 𝑀, we certainly have C ≥ ⟨ Z , Z g ⟩ , and as we now also know the structure of 𝐴, we find that rk ⁡ ( C ∩ A ) = 0 . If rk ⁡ C > 2 , then the group A ⁢ C , would have rank at least five. However, 𝐺 has no rank 5 subgroups, and G = A ⁢ C is impossible by simplicity. Thus rk ⁡ C = 2 and C = ⟨ Z , Z g ⟩ .

We have observed that Z ≰ A , so Z ≰ M g . By the structure of 𝑀 and rank considerations, M g = C G ∘ ⁢ ( Z g ) , so [ Z , Z g ] ≠ 1 . Since 𝐶 has rank 2 and distinct noncommuting rank 1 subgroups, it must be that 𝐶 is nonnilpotent, so we have C / Z ⁢ ( C ) ≅ AGL 1 ⁡ ( L ) for some field 𝐿. This also shows that Z , Z g are isomorphic to tori in AGL 1 ⁡ ( L ) . ∎

Let D = A ∗ C for 𝐴 and 𝐶 as in Claim 2. Set U A = F ∘ ⁢ ( A ) , and let T A be a maximal torus of 𝐴. Similarly define U C and T C . Then A = U A ⋊ T A with Z ⁢ ( A ) ≤ T A and similarly for 𝐶 (see [12, Theorem 2]).

Claim 3

D = F ∘ ⁢ ( D ) ⋊ T , where F ∘ ⁢ ( D ) = U A ⁢ U C and T = T A ⁢ T C .

Proof of claim

We need only show F ∘ ⁢ ( D ) ∩ T = 1 . Let z ∈ F ∘ ⁢ ( D ) ∩ T ≤ Z ⁢ ( D ) . Writing z = a ⁢ c with a ∈ U A , c ∈ U C , we find a ∈ Z ⁢ ( A ) ≤ T A , c ∈ Z ⁢ ( C ) ≤ T C . But U A ∩ T A = 1 = U C ∩ T C , so z = 1 . ∎

Set D ̂ = N G ⁢ ( D ) and Y = G / D ̂ . As 𝐺 has no rank 5 subgroups, D ̂ is maximal in 𝐺. By Claim 1, ( G , Y ) is generically 3-transitive. Let 1 , 2 , 3 ∈ Y be in general position, with 1 representing the trivial coset D ̂ (so G 1 = D ̂ ).

Claim 4

( G , Y ) is 2-transitive. Consequently, rk ⁡ G 1 , 2 = 2 .

Proof of claim

Since 𝑌 is connected of rank 2, we can show the action is 2-transitive by showing that G 1 has a unique orbit of rank 0 (namely { 1 } ) and no orbits of rank 1.

As G 1 is a maximal subgroup, the action is primitive, so { 1 } is indeed the unique rank 0 orbit of G 1 (see for example [31, Lemma 3.4]).

To show G 1 has no orbits of rank 1, it suffices to show it for G 1 ∘ . Assume that G 1 ∘ has an orbit 𝒪 of rank 1; it also has degree 1. Choose y ∈ O , and let 𝐾 be the kernel of the action of G 1 ∘ on 𝒪. Note that G 1 ∘ is generated by its maximal decent tori (using Claim 3), which are conjugate in G 1 ∘ , so 𝐾 does not contain a maximal decent torus of G 1 ∘ . Since 𝐺 has no rank 5 subgroups, ⟨ G 1 ∘ , G y ∘ ⟩ = G , so by simplicity, there is no nontrivial subgroup normalized by both G 1 ∘ and G y ∘ . This then implies that 𝐾 does not contain F ∘ ⁢ ( G 1 ) as otherwise it would imply that F ∘ ⁢ ( G 1 ) = F ∘ ⁢ ( G y ) .

By Hrushovski, 𝐾 has corank 1 or 2 in G 1 ∘ . Combining this with our previous analysis, we find that the corank is equal to 2 (hence rk ⁡ K = 2 ) and that 𝐾 contains a unique rank 1 unipotent subgroup 𝑊 as well as some rank 1 torus. In particular, 𝑊 is characteristic in 𝐾, so normal in G 1 ∘ . We claim that 𝑊 is also normal in G y ∘ , and this will be our final contradiction.

Since G 1 ∘ / K has rank 2, Hrushovski also implies that ( G 1 ∘ ) y / K is a good torus, so ( G 1 ∘ ) y contains a rank 2 good torus 𝑇 of G y ∘ . Since 𝑊 is characteristic in 𝐾, 𝑇 normalizes 𝑊. By the structure of G y ∘ , W ≤ F ⁢ ( G y ∘ ) , implying that F ⁢ ( G y ∘ ) also normalizes 𝑊. But then ⟨ T , F ⁢ ( G y ∘ ) ⟩ = G y ∘ normalizes 𝑊 (as does G 1 ∘ , hence 𝐺), a contradiction. ∎

Claim 5

G 1 = F ∘ ⁢ ( G 1 ) ⋊ G 12 . Thus ( G , Y ) is a split 2-transitive group.

Proof of claim

We begin by showing that G 1 , 2 ∘ is a good torus; assume not.

As in the proof of Claim 4, no nontrivial subgroup is normalized by both G 1 ∘ and G 2 ∘ , so G 1 , 2 ∘ ≠ F ∘ ⁢ ( G 1 ∘ ) . As we are assuming G 1 , 2 ∘ is not a good torus, we find that G 1 , 2 ∘ = W ⁢ S for some rank 1 unipotent group 𝑊 and some rank 1 good torus 𝑆. Also, 𝑊 is normalized by 𝑆 since G 1 , 2 ∘ is solvable.

Consider N : = N G ∘ ( W ) . It contains ⟨ G 1 , 2 ∘ , F ∘ ⁢ ( G 1 ∘ ) , F ∘ ⁢ ( G 2 ∘ ) ⟩ , so rk ⁡ N is at least 4, hence equal to 4. If 𝑁 is nonsolvable, the analysis of Claim 2 could be applied to 𝑁 to force 𝑊 to be a torus, a contradiction. So 𝑁 is solvable, and by Claim 1, 𝑁 is nonnilpotent. Thus F : = F ∘ ( N ) has rank 2 or 3.

First suppose rk ⁡ F = 2 . Then F ∘ ⁢ ( G 1 ∘ ) acts on the factors of 1 ⁢ ⊴ ⁢ W ⁢ ⊴ ⁢ F , which each have rank 1. Any nontrivial action can be linearized, but F ∘ ⁢ ( G 1 ∘ ) has no quotient isomorphic to the multiplicative group of a field. Thus F ∘ ⁢ ( G 1 ∘ ) acts trivially on each factor of the series, implying that F ∘ ⁢ ( G 1 ∘ ) ≤ F . Similarly, F ∘ ⁢ ( G 2 ∘ ) ≤ F , contradicting that rk ⁡ F = 2 .

Next suppose rk ⁡ F = 3 . Since 𝑆 does not commute with F ∘ ⁢ ( G 1 ∘ ) , 𝑆 is not contained in 𝐹, so F = ⟨ F ∘ ⁢ ( G 1 ∘ ) , F ∘ ⁢ ( G 2 ∘ ) ⟩ . Then F ∘ ⁢ ( G 1 ∘ ) is corank 1 in the nilpotent group 𝐹, so 𝐹 normalizes F ∘ ⁢ ( G 1 ∘ ) . Of course, F ∘ ⁢ ( G 1 ∘ ) is normal in G 1 ∘ , and by rank considerations, G 1 ∘ is the connected normalizer in 𝐺 of F ∘ ⁢ ( G 1 ∘ ) . Thus F ≤ G 1 ∘ . However, we know the structure of 𝐹 (unipotent of rank 3) and of G 1 ∘ (rank 4 containing a rank 2 good torus), so we have a contradiction.

Thus G 1 , 2 ∘ is a good torus. We may apply [31, Lemma 3.7] to see that G 1 and G 12 are connected, and then Claim 3 completes the proof. ∎

Claim 6

Contradiction.

Proof of claim

We appeal to the theory of split 2-transitive permutation groups (via the theory of Moufang sets). Since both factors in the splitting

G 1 = F ∘ ⁢ ( G 1 ) ⋊ G 12

are abelian, we get G ≅ PSL 2 ⁡ ( F ) (see for example [30, Corollary 1.2]). However, proper definable subgroups of PSL 2 ⁡ ( F ) are solvable while 𝑀 is not. ∎

This completes the proof of Proposition 3. ∎

3 Proof of the main result

Throughout this final section, we adopt the hypotheses of the Theorem as well as the additional notation presented in Section 1.3. In particular, ( G , X ) is a generically sharply 𝑡-transitive permutation group of finite Morley rank, ( 1 , … , t ) ∈ X t is a generic 𝑡-tuple, and r = rk ⁡ X . By existing work of Hrushovski (see [9, Theorem 11.98]) and the present authors [2], we may assume r ≥ 3 .

We begin by addressing connectedness of point-stabilizers, which we use in our analysis of G [ t − 1 ] and G [ t − 2 ] .

Lemma 3.1

If t ≥ 2 , then for every 0 ≤ m < t , the pointwise stabilizer G [ m ] of 1 , … , m is connected of rank ( t − m ) ⋅ r , where we take G [ 0 ] to be 𝐺.

Proof

Since t ≥ 2 , by [7, Lemma 1.8 (3)], the set 𝑋 is connected; this is a key point.

Now, as the action of 𝐺 is generically sharply 𝑡-transitive, G [ m ] acts generically sharply ( t − m ) -transitively, so if O ⊆ X t − m is the generic orbit of G [ m ] , then G [ m ] is in definable bijection with 𝒪 (making essential use of sharpness). Since 𝒪 is generic in X t − m , 𝒪 has rank ( t − m ) ⋅ r as well as degree 1 by our initial observation, so the same is true of G [ m ] . ∎

As indicated in Section 1.3, Proposition 1 will reduce our work to consideration of two cases: (1) G [ t − 1 ] is solvable or (2) r = 3 with G [ t − 1 ] ≅ PGL 2 ⁡ ( F ) . We treat these two cases first and then our eventual proof of the Theorem will simply tie things together. We begin with the solvable case – this falls quickly thanks to Proposition 2.

Proposition 4

If G [ t − 1 ] is solvable, then t ≤ r + 2 .

Proof

Assume G [ t − 1 ] is solvable and, towards a contradiction, that t ≥ r + 3 . Then Σ t ≅ Sym ⁡ ( t − 1 ) ≥ Sym ⁡ ( r + 2 ) , and we consider the faithful action of Σ t on the rank 𝑟 group G [ t − 1 ] (see Section 1.3).

By [4, Lemma 2.7], G [ t − 1 ] is abelian, and then [14] takes over – via the First Geometrisation Lemma followed by the Recognition Lemma – to show that G [ t − 1 ] is an elementary abelian 𝑝-group for some 𝑝 dividing t − 1 . However, since G [ t − 1 ] is abelian, Proposition 2 applies to the action of G [ t − 2 ] on its generic orbit, and we find that G [ t − 1 ] has a quotient isomorphic to L × for some algebraically closed field 𝐿, a contradiction. ∎

We now address the exceptional case of r = 3 and G [ t − 1 ] ≅ PGL 2 ⁡ ( F ) . The goal is to eliminate this possibility, which we phrase as follows.

Proposition 5

If r = 3 and t ≥ 5 , then G [ t − 1 ] is solvable.

Our proof of Proposition 5 will take some preparation.

Lemma 3.2

Assume r = 3 , t ≥ 5 , and G [ t − 1 ] is nonsolvable. Then the following hold:

G [ t − 1 ] ≅ PSL 2 ⁡ ( K ) for some rank 1 field 𝐾, with S t acting by inner automorphisms;
G [ t − 1 ] is a maximal connected definable subgroup of G [ t − 2 ] ;
G [ t − 2 ] is quasisimple with finite center.

Proof

Set B = G [ t − 2 ] and H = G [ t − 1 ] ; recall from Lemma 3.1 that they are connected and of ranks 6 and 3 respectively.

Given the classification of groups of rank 3 (see [12, 18]) and the faithful action S t ≥ Sym ⁡ ( 4 ) on 𝐻, we have that H ≅ PGL 2 ⁡ ( K ) for some algebraically closed field 𝐾, with S t acting by inner automorphisms.

We next show that 𝐵 is quasisimple with finite center. Suppose 𝑁 is a proper connected normal subgroup of 𝐵 and of maximal rank with respect to those properties; we aim to show N = 1 . Since 𝐻 is simple, H ∩ N = 1 , so rk ⁡ N ≤ 3 (since rk ⁡ B = 6 ). Now, B / N is nonsolvable (as it embeds 𝐻), so by maximality of 𝑁, B / N is quasisimple with finite center. And since there are no quasisimple groups of rank 4 or 5 that have involutions [31, 16], rk ⁡ B / N = rk ⁡ N = 3 .

We now have H ⁢ N = B and N ∩ H = 1 . Thus 𝑁 acts regularly on the generic orbit of 𝐵 on 𝑋, and the action of 𝐻 on this orbit is isomorphic to the action of 𝐻 on 𝑁 by conjugation. Thus 𝐻 acts generically transitively on 𝑁 by conjugation. If 𝑁 is solvable, then 𝑁 is abelian. But then [8] implies that the action of 𝐻 on 𝑁 is the adjoint action, and this cannot be generically transitive since 𝐻 preserves the determinant. Otherwise, if 𝑁 is nonsolvable, then 𝑁 is quasisimple of rank 3, so of the form ( P ) ⁢ SL . However, no such group supports a generically transitive automorphism group. Thus rk ⁡ N ≠ 3 , so 𝐵 is indeed quasisimple with finite center.

Finally, we show that 𝐻 is a maximal connected definable subgroup of 𝐵. Indeed, if 𝑀 is a proper connected definable subgroup of 𝐵 that properly contains 𝐻, then 𝑀 is nonsolvable and rk ⁡ M ≥ 4 . Moving to the quotient of 𝐵 by its center, we have a contradiction to Proposition 3. ∎

Our contradictory assumption that G [ t − 1 ] is nonsolvable forces G [ t − 2 ] to be quasisimple with a finite center and many involutions (due to the structure of G [ t − 1 ] ), and this creates serious tension with the Algebraicity Conjecture. Since G [ t − 2 ] has rank 6, the conjecture implies (as seen in Lemma 3.3 below) that G [ t − 2 ] should be of the form PSL 2 so should certainly not have a subgroup like G [ t − 1 ] . This is exactly our contradiction when G [ t − 2 ] is of even type thanks to [1]. In odd type, nonalgebraic G [ t − 2 ] remains a possibility, but we are able to sufficiently pin down its structure – this time thanks to Proposition 3 and [15] – to again reach a contradiction.

Lemma 3.3

The only infinite simple algebraic groups of Morley rank at most 6 are PSL 2 ⁡ ( K ) over a field 𝐾 of Morley rank 1 or 2.

Proof

Let 𝐺 be an infinite simple algebraic group over a field 𝐾 of finite Morley rank. By a theorem of Macintyre, 𝐾 is algebraically closed, and thus the structure theory of simple algebraic groups over algebraically closed fields applies. According to this, the dimension of 𝐺 over the pure field structure is 2 ⁢ u + l , where 𝑢 is the dimension of the unipotent subgroup generated by a fixed set of positive roots and 𝑙 is the Lie rank of 𝐺 (see for example [24, Section 28.5]). Moreover, u ≥ l ≥ 1 . It follows that rk ⁡ ( G ) = ( 2 ⁢ u + l ) ⁢ rk ⁡ ( K ) ≥ 3 ⁢ rk ⁡ ( K ) ≥ 3 . If l > 1 , then the smallest value of 𝑢 is 3 (see for example [23, Section 9.3]). Thus l = 1 since rk ⁡ ( G ) ≤ 6 . In this case, the only possibility for 𝐺 is PSL 2 ⁡ ( K ) (see [24, Corollary 32.3]), and rk ⁡ ( K ) is either 1 or 2. ∎

Corollary 3.4

Under the assumptions of Lemma 3.2, G [ t − 2 ] is of odd type.

Proof

Set B = G [ t − 2 ] . Then 𝐵 is quasisimple with finite center according to Lemma 3.2 3. If 𝐵 contains a nontrivial unipotent 2-subgroup, so does B / Z ⁢ ( B ) . By the classification of the simple groups of finite Morley rank of even and mixed type [1], B / Z ⁢ ( B ) is a simple linear algebraic group over an algebraically closed field of characteristic 2. Lemma 3.3 then implies B / Z ⁢ ( B ) ≅ PSL 2 ⁡ ( K ) with 𝐾 algebraically closed, but G [ t − 1 ] is not solvable by Lemma 3.2 1, violating the structure of PSL 2 ⁡ ( K ) . ∎

The tension in odd type remains. Things will begin to give as we invoke Proposition 3 and ultimately Fact 2.10 to show that if G [ t − 2 ] is not of the form PSL 2 , then it is still – in some sense – close (being of type CiBo₂). In any case, the structure of G [ t − 2 ] becomes rather transparent and, with a little more effort, quickly falls apart.

The next lemma provides the bridge from Proposition 3 to Fact 2.10; the proof of Proposition 5 will follow.

Lemma 3.5

Under the assumptions of Lemma 3.2, C G [ t − 2 ] ∘ ⁢ ( i ) is solvable for each involution i ∈ G [ t − 2 ] .

Proof

Let B = G [ t − 2 ] and C = C B ∘ ⁢ ( i ) . By [7, Proposition 5.15], i ∈ C . Suppose now that 𝐶 is not solvable. By Proposition 3, rk ⁡ C ≤ 3 ; this is a crucial point that eliminates the possibility of 𝐶 containing a copy of PSL 2 .

The classification of nonsolvable groups of rank at most 3 (and that i ∈ C ) implies that C ≅ SL 2 ⁡ ( F ) with 𝐹 of characteristic not 2. But then the Sylow 2-subgroup of 𝐵 matches that of SL 2 ⁡ ( F ) , contrary to the fact that the Sylow 2-subgroup of G [ t − 1 ] ≤ B matches that of PSL 2 ⁡ ( F ) by Lemma 3.2 1. ∎

Proof of Proposition 5

We work with a counterexample; assume r = 3 , t ≥ 5 , and H = G [ t − 1 ] is nonsolvable. Set B = G [ t − 2 ] . By Lemma 3.2 and Corollary 3.4 together with Proposition 3, 𝐵 is an N ∘ ∘ -group of odd type, and by Lemma 3.5, we may apply Fact 2.10. The presence and structure of 𝐻 given in Lemma 3.2 implies that we are in type CiBo₂.

Consider C = C B ∘ ⁢ ( i ) for any i ∈ I ⁢ ( B ) . Then 𝐶 is an abelian Borel subgroup of 𝐵 and rk ⁡ C = 2 . We claim that 𝐶 is maximal connected in 𝐵. Indeed, if not, then C < M with 𝑀 connected of rank 3 or 4, and as 𝐶 is a Borel subgroup of 𝐵, 𝑀 is nonsolvable. By Proposition 3, 𝑀 must have rank 3, but no nonsolvable rank 3 group has a rank 2 abelian subgroup. Thus 𝐶 is indeed maximal connected in 𝐵.

Next, notice that if i ∈ H , then T : = ( C ∩ H ) ∘ is a rank one good torus, which is the definable closure of a 2-torus in 𝐻. In particular, 𝑇 is a characteristic rank 1 subgroup of 𝐶. And as involutions are conjugate in 𝐵 (as a consequence of having type CiBo₂), C B ∘ ⁢ ( i ) has such a characteristic rank 1 subgroup for every choice of i ∈ I ⁢ ( B ) .

We apply our observations to C = C B ∘ ⁢ ( ω ) for ω = ( t − 1 , t ) . Note that H ω ∩ H = 1 , so H ∩ C = 1 . We study the action of Σ t − 1 , t ≅ Sym ⁡ ( t − 2 ) ≥ Sym ⁡ ( 3 ) on 𝐶 and on 𝐻 to show, for a 3-cycle γ ∈ Σ t − 1 , t , that C B ∘ ⁢ ( γ ) contains 𝐶 and has rank at least 3, contradicting the fact that 𝐶 is maximal connected in 𝐵.

Let 𝑇 be the definable closure of a 2-torus in 𝐶. Then Σ t − 1 , t acts on the rank 1 factors of the series 1 ≤ T ≤ C . Thus [ γ , C ] ≤ T , and [ γ , γ , C ] = 1 . If [ γ , C ] ≠ 1 , then (by the usual quadratic arguments) [ γ , C ] is an elementary abelian 3-group, which also must be equal to 𝑇, a contradiction. We conclude that [ γ , C ] = 1 , so C ≤ C B ∘ ⁢ ( γ ) .

By Lemma 3.2 1, 𝛾 acts on 𝐻 as an inner automorphism, so 𝛾 centralizes a rank 1 connected subgroup 𝐴 of 𝐻. As observed above, H ∩ C = 1 , so ⟨ C , A ⟩ is a connected subgroup of C B ∘ ⁢ ( γ ) of rank at least 3. Additionally, Σ t − 1 , t acts faithfully on 𝐵 (see Section 1.3), so C B ∘ ⁢ ( γ ) is a proper subgroup of 𝐵 that properly contains 𝐶, a final contradiction. ∎

Now, we complete the proof of the main theorem of the paper.

Proof of the Theorem

We work towards a contradiction under the assumption that t ≥ r + 3 . As we noted at the beginning of this section, we may assume that r ≥ 3 . Let H = G [ t − 1 ] . By Lemma 3.1, 𝐻 is connected of Morley rank 𝑟.

By Proposition 4, 𝐻 is not solvable, and then invoking Proposition 5, we have that r ≥ 4 . We have seen that 𝐻 is connected of rank 𝑟 and Σ t ≅ Sym ⁡ ( t − 1 ) acts faithfully on 𝐻. As t − 1 ≥ r + 2 ≥ 6 , Proposition 1 implies t − 1 ≤ rk ⁡ ( H ) = r , a contradiction. ∎

Funding source: National Science Foundation

Award Identifier / Grant number: DMS-1954127

Funding statement: The work of the second author was partially supported by the National Science Foundation under grant No. DMS-1954127.

Acknowledgements

We warmly thank the anonymous referee for a careful reading of the article and numerous helpful suggestions. Among other things, their remarks led us to clarify a number of points and to identify a small case that was overlooked in an earlier version of Proposition 1.

Communicated by: Michael Giudici

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Received: 2024-07-12

Revised: 2025-01-15

Published Online: 2025-03-04

Published in Print: 2025-11-01

Articles in the same Issue

https://doi.org/10.1515/jgth-2024-0152