Removing the L-group hypothesis: bounding Alt(n)-actions and the degree of generic sharp transitivity for permutation groups of finite Morley rank

Altınel and Wiscons (J. Group Theory 28 (2025) 1261-1284) proved that a generically sharply $t$-transitive permutation group of finite Morley rank on a set of rank $r$ satisfies $t \le r+2$, provided the pointwise stabilizer $H$ of a generic $(t-1)$-tuple is an $L$-group, i.e. a group whose definable simple sections of odd type are all algebraic over algebraically closed fields. The hypothesis enters through a single ingredient, their Proposition 1: if $\operatorname{Alt}(n)$ or $\operatorname{Sym}(n)$ acts definably and faithfully by automorphisms on a nonsolvable connected $L$-group $H$ of finite Morley rank, then $n \le \operatorname{rk}H$ for $n \ge 9$. The authors state that removing the $L$-hypothesis from Proposition 1 would remove it from the main theorem, but despite effort they could not achieve this. The resulting problem: prove the bound $n \le \operatorname{rk}H$ (for all sufficiently large $n$) for definable faithful $\operatorname{Alt}(n)$/$\operatorname{Sym}(n)$-actions on arbitrary nonsolvable connected groups of finite Morley rank — the problem posed by Corredor–Deloro–Wiscons (2023) — and thereby the unconditional bound $t \le r+2$. No progress removing the hypothesis is known: the underlying obstruction, the missing structure theory of odd-type simple groups of finite Morley rank, remains open according to Tent's 2026 ICM survey.

Background

Morley rank is the model-theoretic analogue of algebraic dimension: an integer-valued rank on the definable sets of a stable first-order theory, and a group of finite Morley rank is a group definable in such a theory. These groups behave like algebraic groups in many respects — indeed, by the Cherlin–Zilber Algebraicity Conjecture every infinite simple group of finite Morley rank should be isomorphic to an algebraic group over an algebraically closed field — but the conjecture is confirmed only in rank at most 3 and in even type, and remains wide open in general. Throughout, 'definable' (including interpretable) plays the role that 'constructible' plays for algebraic groups, and $\operatorname{rk}$ denotes Morley rank.

A permutation group $(G, X)$ of finite Morley rank is a faithful, definable action of a definable group $G$ on a definable set $X$. It is generically $t$-transitive if $G$ has an orbit $O$ on $X^t$ with $\operatorname{rk}(X^t \setminus O) < \operatorname{rk}(X^t)$ — transitivity on tuples outside a lower-dimensional exceptional set — and generically sharply $t$-transitive if, in addition, no nontrivial element of $G$ fixes any tuple of that orbit pointwise. The maximum such $t$ is the degree of generic transitivity. Unlike ordinary $t$-transitivity, the generic notion admits large degrees: $\operatorname{PGL}_{r+1}(F)$ (the projective general linear group) acting on projective space $\mathbb{P}^r(F)$ is generically sharply $(r+2)$-transitive, since the projective bases form a single orbit and only the diagonal torus fixes one pointwise.

Borovik and Cherlin proved that $\operatorname{rk}G$ is bounded in terms of $r = \operatorname{rk}X$ and posed the Main Problem (their Problem 9): if $(G, X)$ is a transitive, generically $t$-transitive permutation group of finite Morley rank with $t \ge r+2$, then $(G, X)$ is $\operatorname{PGL}_{r+1}(F)$ acting on $\mathbb{P}^r(F)$, so $t = r+2$. The problem divides into Problem 1 (Sharpness: $t \ge r+2$ forces generic sharp $t$-transitivity), Problem 2 (Bound: generically sharply $t$-transitive implies $t \le r+2$), and Problem 3 (Identification: $t = r+2$ forces the projective-linear model). The Main Problem is known only in rank $\le 2$ (Altınel–Wiscons, 2018), in algebraically closed fields of characteristic zero (Freitag–Moosa, building on Popov's classification of generically multiply transitive actions of algebraic groups), in differentially closed fields (Freitag–Jimenez–Moosa, 2025), and for actions on modules (Berkman–Borovik).

For a group $H$ of finite Morley rank, the interaction with the Algebraicity Conjecture is organized by involutions (elements of order 2): $H$ has degenerate type (no involutions), even type (an infinite elementary abelian 2-group but no Prüfer 2-group $C_{2^\infty}$, the quasicyclic 2-group), odd type (a Prüfer 2-group but no infinite elementary abelian 2-group), or mixed type; every connected group of finite Morley rank has exactly one of these types, and an infinite simple one cannot have mixed type. Even-type simple groups are classified (they are algebraic); degenerate type constitutes a Feit–Thompson-type barrier with little progress; odd type is understood only conditionally. An $L$-group is a group of finite Morley rank all of whose definable simple sections (quotients of definable subgroups) of odd type are algebraic groups over algebraically closed fields — a hypothesis designed to exclude the conjecturally nonexistent non-algebraic odd-type simple groups while assuming nothing about degenerate-type sections.

Altınel–Wiscons (2025) proved Problem 2 in all ranks under the $L$-hypothesis: if $(G, X)$ is generically sharply $t$-transitive of finite Morley rank, $\operatorname{rk}X = r$, and $H$, the pointwise stabilizer of a generic $(t-1)$-tuple, is an $L$-group — automatic when $H$ is solvable, when $r \le 5$, or when $G$ has even type — then $t \le r+2$. The engine is Proposition 1: if $\operatorname{Alt}(n)$ or $\operatorname{Sym}(n)$ (the alternating or symmetric group) acts definably and faithfully by automorphisms on a nonsolvable connected $L$-group $H$ of finite Morley rank, then $n \le \operatorname{rk}H$ for $n \ge 9$, with one exception ($n = 5$ and $H \cong Z(H) \times \operatorname{PGL}_2(F)$ with $\operatorname{char}F = 5$). Proposition 1 reduces to the purely algebraic Corollary 2.6: a definable faithful $\operatorname{Alt}(n)$- or $\operatorname{Sym}(n)$-action on a simple algebraic group $G$ satisfies $n \le \dim G$ for $n \ge 9$, via Wagner's classical bounds on minimal faithful representation degrees of alternating groups. The degenerate-type case of the $\operatorname{Alt}(n)$-action problem was settled by the same authors (2024), and the solvable case carries the weaker bound $n \le \operatorname{rk}H + 2$. Corredor–Deloro–Wiscons (2023), whose 'modules with an additive dimension' axiomatize representation theory simultaneously in the finite, o-minimal, and finite Morley rank settings, solved the abelian case and posed the nonsolvable problem explicitly. Altınel–Wiscons state that the $L$-hypothesis enters their theorem only through Proposition 1, so removing it from Proposition 1 would remove it from the theorem; despite effort they could not extend Corollary 2.6 from simple algebraic groups to simple odd-type groups of finite Morley rank.

Problem Statement

Remove the $L$-group hypothesis from the rank-free bound on generic sharp transitivity; that is, prove the same bounds for arbitrary nonsolvable connected groups of finite Morley rank, in which non-algebraic odd-type definable simple sections are not excluded. Two linked formulations; the first implies the second by the source paper's own reduction.

(1) The Corredor–Deloro–Wiscons problem (Proposition 1 without the $L$-hypothesis). Suppose $\operatorname{Alt}(n)$ or $\operatorname{Sym}(n)$ acts definably and faithfully by automorphisms on a nonsolvable connected group $H$ of finite Morley rank. Prove that $n \le \operatorname{rk}H$ for all sufficiently large $n$, with no assumption on the definable simple sections of $H$. (Known: $n \ge 9$ when $H$ is an $L$-group, with the single exception $n = 5$ and $H \cong Z(H) \times \operatorname{PGL}_2(F)$ with $\operatorname{char}F = 5$; the bound also holds when $H$ has degenerate type, i.e., no involutions.)

(2) The Main Theorem without the $L$-hypothesis (Borovik–Cherlin Problem 2, unconditional). Prove that every generically sharply $t$-transitive permutation group $(G, X)$ of finite Morley rank with $\operatorname{rk}X = r$ satisfies $t \le r+2$, with no hypothesis on the pointwise stabilizer $H$ of a generic $(t-1)$-tuple — in particular not that $H$ is an $L$-group, solvable, or that $r \le 5$.

Altınel–Wiscons identify the key missing ingredient for (1): an extension of their Corollary 2.6 — for $n \ge 9$, $n \le \dim G$ for definable faithful $\operatorname{Alt}(n)$/$\operatorname{Sym}(n)$-actions on simple algebraic groups — to simple groups of finite Morley rank of odd type.

References

1. Tuna Altınel and Joshua Wiscons, 'Bounding the degree of generic sharp transitivity', Journal of Group Theory 28 (2025), no. 6, 1261-1284, DOI 10.1515/jgth-2024-0152 (arXiv:2407.09636). https://doi.org/10.1515/jgth-2024-0152
2. Luis Jaime Corredor, Adrien Deloro, and Joshua Wiscons, 'Sym(n)- and Alt(n)-modules with an additive dimension', Journal of Algebra 623 (2023) 1-33, DOI 10.1016/j.jalgebra.2023.02.009 (arXiv:2111.11498). https://doi.org/10.1016/j.jalgebra.2023.02.009
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13. Katrin Tent, 'From the Cherlin-Zilber Conjecture via sharply 2-transitive groups to the Burnside problem', arXiv:2606.18207 (2026), prepared for the Proceedings of the ICM 2026. https://arxiv.org/abs/2606.18207
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