The Cherlin–Zilber algebraicity conjecture in odd type: simple groups of finite Morley rank with normal 2-rank at most 2

The Cherlin–Zilber algebraicity conjecture predicts that every infinite simple group of finite Morley rank — a group whose definable sets all carry a finite model-theoretic dimension, the natural generalization of an algebraic group over an algebraically closed field — is itself a simple algebraic group over an algebraically closed field. The conjecture is settled for groups of even and mixed type (Sylow 2-subgroups of infinite 2-rank) and for Morley rank at most 3. For groups of odd type, the trichotomy theorem of Borovik and Burdges (2008), combined with the generic identification, strong embedding and minimal simple theorems of the surrounding program, reduced the problem to groups whose normal 2-rank — which coincides with the Prüfer 2-rank for connected groups by a lemma of Deloro (2012) — is at most 2: the thin and quasithin cases, with expected algebraic targets $PSL_2$, $PSL_3$, $PSp_4$ and $G_2$. This residual case remains open. Cherlin's 2024 survey describes the quasithin 2-rank 2 case (target $PSL_3$) as 'largely unexplored'; the quasithin results beyond 2-rank 2 exist only as unpublished drafts conditional on the conjectural NTA2 condition; and Tent's 2026 survey reports the overall conjecture 'still wide open', with nonsplit sharply 2-transitive groups of finite Morley rank as the leading candidate source of counterexamples. The problem asks for a proof that every simple group of finite Morley rank of odd type with normal 2-rank at most 2 is a simple algebraic group over an algebraically closed field of characteristic $\neq 2$, or for an explicit nonalgebraic counterexample.

Background

Morley rank is the model-theoretic analogue of algebraic dimension: an ordinal-valued rank assigned to the definable sets (sets cut out by a first-order formula, possibly with parameters) of a structure, invariant under definable bijections. In an algebraically closed field it is finite and agrees with Zariski dimension, so every algebraic group over an algebraically closed field is a group of finite Morley rank: a group whose definable sets all have finite Morley rank. The Cherlin–Zilber algebraicity conjecture, formulated around 1979, inverts this observation: every infinite simple group of finite Morley rank should be isomorphic to a simple algebraic group over an algebraically closed field. It is the central open problem of the model theory of groups, and this problem lies within the Borovik program, a planned classification of the would-be counterexamples modeled on the classification of finite simple groups.

The program's first cut is by 2-structure, through the Sylow 2-subgroups (maximal 2-subgroups) of the group. These divide simple groups of finite Morley rank into four types: degenerate (no involutions, i.e. no elements of order 2), even and mixed (Sylow 2-subgroups containing an infinite elementary abelian 2-subgroup — in even type the exponents of 2-elements are bounded, in mixed type they are not and a nontrivial divisible abelian 2-subgroup is present), and odd (Sylow 2-subgroups that are an extension of a divisible abelian 2-group by a finite group). The monograph of Altınel, Borovik and Cherlin (2008) settled the even and mixed types: such simple groups are algebraic. The degenerate type is entangled with the still-mysterious 'bad groups' (the minimal connected simple groups of degenerate type) and lies outside 2-structure methods. Odd type is the arena of this problem.

Two numerical invariants measure the 2-structure in odd type. A 2-torus is a divisible abelian 2-group, a direct product of copies of the Prüfer 2-group (the group of all dyadic rationals modulo the integers); the Prüfer 2-rank of the group is the number of copies in a maximal 2-torus. It is the analogue of Lie rank: a Chevalley group (one of the universal algebraic groups: $PSL_n$, $PSp_{2n}$, $G_2$, and so on) of Lie rank $\ell$ over an algebraically closed field of characteristic $\neq 2$ has Prüfer 2-rank $\ell$. The normal 2-rank is the largest rank of an elementary abelian 2-subgroup (an abelian group of exponent 2) that is normal in some Sylow 2-subgroup. Deloro proved in 2012 that for connected groups of finite Morley rank and finite 2-rank the two invariants coincide, so the modern subdivision of odd type — thin (Prüfer 2-rank 1), quasithin (Prüfer 2-rank 2), generic (at least 3) — is at the same time a subdivision by normal 2-rank.

The engine of the program is induction through $K^*$-groups: groups of finite Morley rank whose proper definable simple sections (a section is a quotient of a definable subgroup by a definable normal subgroup) are all algebraic; a minimal counterexample to the conjecture is automatically a $K^*$-group. Two families of configurations must be controlled. The first is the uniqueness case, detected by the weak 2-generated core $\Gamma^{0}_{S,2}(G)$: the definable hull (smallest definable subgroup containing it) of the subgroup generated by the normalizers, in a Sylow 2-subgroup $S$, of those involution four-groups $U \cong (\mathbb{Z}/2)^2$ whose centralizer $C_S(U)$ has 2-rank greater than 2. If this core is a proper subgroup, the analysis funnels into strongly embedded subgroups — proper definable subgroups containing the centralizer of each of their involutions — a rigid configuration that forces severe rank restrictions, and into minimal connected simple groups (connected simple groups of finite Morley rank all of whose proper definable connected subgroups are solvable), which form the base case of the induction. The second is the identification case, in which $G$ is recognized as a Chevalley group, in odd type necessarily over an algebraically closed field of characteristic $\neq 2$. The main technique for both, borrowed from finite group theory, is signalizer functor theory: a machinery for balancing the components (quasisimple subnormal definable subgroups) that arise in centralizers of involutions.

Borovik and Burdges (2008) prove a tameness-free trichotomy theorem: a simple $K^*$-group of finite Morley rank and odd type with normal 2-rank at least 3 — or, more generally, with an eight-group (an elementary abelian 2-group of order 8) centralizing the maximal 2-torus — either has a proper weak 2-generated core or is algebraic. Combined with the Strong Embedding Theorem of Borovik–Burdges–Nesin (a proper 2-generated core of such a group makes it a minimal connected simple group carrying a strongly embedded subgroup) and the Minimal Simple Theorem of Burdges–Cherlin–Jaligot (a minimal connected simple group of odd type with a strongly embedded subgroup has Prüfer 2-rank one), the resulting 'Status' corollary reads: a simple $K^*$-group of finite Morley rank and odd type with normal 2-rank at least 3 and Prüfer 2-rank at least 2 is algebraic. What survives is exactly the small-rank regime: normal (equivalently Prüfer) 2-rank at most 2, where the expected algebraic outcomes are $PSL_2$ in the thin case, and $PSL_3$, $PSp_4$ or $G_2$ in the quasithin case — of ordinary 2-ranks (the maximal rank of an elementary abelian 2-subgroup) 2, 4 and 3 respectively. Extending the induction from $K^*$ to $L^*$-groups — where proper definable connected simple sections may also be of degenerate type — requires in odd type an additional hypothesis, the NTA2 condition (no definable action of a 2-torus on a degenerate-type section), which itself remains conjectural.

In the refutation direction, Tent's 2026 survey presents the nonsplit sharply 2-transitive groups of finite Morley rank — permutation groups in which exactly one element carries any ordered pair of distinct points to any other such pair, and which have no nontrivial abelian normal subgroup — as the leading candidate source of counterexamples to the conjecture.

Problem Statement

Let $G$ be an infinite simple group of finite Morley rank (a group whose definable sets all carry a finite model-theoretic dimension) of odd type — its Sylow 2-subgroups (maximal 2-subgroups) are an extension of a divisible abelian 2-group by a finite group — with normal 2-rank at most 2, where the normal 2-rank is the maximal rank of an elementary abelian 2-subgroup normal in a Sylow 2-subgroup. Since an infinite simple group of finite Morley rank is connected, Deloro's coincidence lemma (2012) makes this equivalent to Prüfer 2-rank at most 2 — a maximal 2-torus (maximal divisible abelian 2-subgroup) containing at most two copies of the Prüfer 2-group (the group of dyadic rationals modulo the integers): the thin case (one copy) or the quasithin case (two copies). Decide algebraicity in this remaining regime of the Cherlin–Zilber algebraicity conjecture.

(A) Prove that $G$ is isomorphic to one of the Chevalley groups (the universal algebraic groups of Lie theory) $PSL_2$, $PSL_3$, $PSp_4$ or $G_2$ over an algebraically closed field of characteristic $\neq 2$ — in the thin case $PSL_2$; in the quasithin case $PSL_3$, $PSp_4$ or $G_2$ according as the ordinary 2-rank of $G$ (the maximal rank of an elementary abelian 2-subgroup) is 2, at least 4 or 3. Together with the settled case of normal 2-rank at least 3 (the 'Status' corollary of Borovik–Burdges 2008), this would complete the odd-type case of the Cherlin–Zilber algebraicity conjecture.

(B) Or refute the conjecture in this regime by constructing such a group $G$ that is not isomorphic to any algebraic group over an algebraically closed field.

Concrete open configurations within (A), as isolated by the surveyed literature: the quasithin 2-rank 2 case aiming at $PSL_3$, currently largely unexplored; the quasithin 2-rank 3 case, where the expected outcome is $G_2$ but an exotic characteristic-3 configuration (all involutions conjugate, involution centralizer of the form $SL_3 * SL_3$ over fields of characteristic 3) must be eliminated or realized; the thin case, which passes through the theory of minimal connected simple groups of odd type (connected simple groups all of whose proper definable connected subgroups are solvable; $PSL_2$ is the prototype) and of Moufang sets (the doubly transitive permutation structures equivalent to split BN-pairs of rank 1); and the NTA2 condition itself, which remains conjectural.

References

1. Alexandre V. Borovik and Jeffrey Burdges, 'A new trichotomy theorem for groups of finite Morley rank', Journal of the London Mathematical Society 77 (2008), 1–14. DOI: 10.1112/jlms/jdm088. https://doi.org/10.1112/jlms/jdm088 (full text also at arXiv:0711.4169, https://arxiv.org/abs/0711.4169)
2. Alexandre V. Borovik, Jeffrey Burdges, and Ali Nesin, 'Uniqueness cases in odd-type groups of finite Morley rank', Journal of the London Mathematical Society 77 (2008), 240–252. DOI: 10.1112/jlms/jdm106. https://doi.org/10.1112/jlms/jdm106
3. Jeffrey Burdges, 'Signalizers and balance in groups of finite Morley rank', Journal of Algebra 321 (2009), 1383–1406. DOI: 10.1016/j.jalgebra.2008.12.001. https://doi.org/10.1016/j.jalgebra.2008.12.001 (also arXiv:0711.4210, https://arxiv.org/abs/0711.4210)
4. Jeffrey Burdges, Gregory Cherlin, and Eric Jaligot, 'Minimal connected simple groups of finite Morley rank with strongly embedded subgroups', Journal of Algebra 314 (2007), 581–612. DOI: 10.1016/j.jalgebra.2006.12.031. https://doi.org/10.1016/j.jalgebra.2006.12.031
5. Ayşe Berkman, Alexandre V. Borovik, Jeffrey Burdges, and Gregory Cherlin, 'A generic identification theorem for L*-groups of finite Morley rank', Journal of Algebra 319 (2008), 50–76. DOI: 10.1016/j.jalgebra.2007.10.012. https://doi.org/10.1016/j.jalgebra.2007.10.012
6. Adrien Deloro, 'p-Rank and p-groups in algebraic groups', Turkish Journal of Mathematics 36 (2012), 578–582. DOI: 10.3906/mat-1103-52. https://doi.org/10.3906/mat-1103-52
7. Joshua Wiscons, 'Moufang sets of finite Morley rank of odd type', Journal of Algebra 402 (2014), 479–498. DOI: 10.1016/j.jalgebra.2014.01.001. https://doi.org/10.1016/j.jalgebra.2014.01.001
8. Olivier Frécon, 'Simple groups of Morley rank 3 are algebraic', Journal of the American Mathematical Society 31 (2018), 643–659. DOI: 10.1090/jams/892. https://doi.org/10.1090/jams/892
9. Gregory Cherlin, 'Around the algebraicity problem in odd type', Model Theory 3 (2024), 505–538. DOI: 10.2140/mt.2024.3.505. https://doi.org/10.2140/mt.2024.3.505 (full text also at https://msp.org/mt/2024/3-2/mt-v3-n2-p13-s.pdf)
10. Katrin Tent, 'From the Cherlin-Zilber Conjecture via sharply 2-transitive groups to the Burnside problem', preprint (2026). arXiv:2606.18207. https://arxiv.org/abs/2606.18207
11. Ayşe Berkman and Alexandre V. Borovik, 'The Şiş Kebap Theorem: A Generic Identification Theorem for Groups of Finite Morley Rank', preprint (2026). arXiv:2608.01161. https://arxiv.org/abs/2608.01161
12. Tuna Altınel, Alexandre Borovik, and Gregory Cherlin, 'Simple Groups of Finite Morley Rank', Mathematical Surveys and Monographs 145, American Mathematical Society (2008). ISBN 978-0-8218-4305-5. DOI: 10.1090/surv/145. https://doi.org/10.1090/surv/145
13. Ayşe Berkman and Alexandre V. Borovik, 'Groups of finite Morley rank with a generically sharply multiply transitive action', Journal of Algebra 513 (2018), 113–132. DOI: 10.1016/j.jalgebra.2018.07.033. https://doi.org/10.1016/j.jalgebra.2018.07.033
14. Ayşe Berkman and Alexandre V. Borovik, 'Primitive permutation groups of finite Morley rank and affine type', Contemporary Mathematics 830 (2025), 31–51. DOI: 10.1090/conm/830/16557. https://doi.org/10.1090/conm/830/16557
15. Jeffrey Burdges and Gregory Cherlin, 'L*-Groups of odd type with restricted 2-toral automorphisms, I–IV', unpublished drafts (2022), listed at https://sites.math.rutgers.edu/~cherlin/Paper/inprep.html; contents described in G. Cherlin, 'Around the algebraicity problem in odd type', Model Theory 3 (2024), 505–538, DOI: 10.2140/mt.2024.3.505