The uniqueness case for odd-type groups of finite Morley rank: strong embedding and quasithin classification in the tameness-free $L^*$-theory

Open problem originating from Borovik–Burdges–Nesin, "Uniqueness cases in odd-type groups of finite Morley rank" (2008), a step in the Borovik program toward the Cherlin–Zilber algebraicity conjecture that every simple group of finite Morley rank is a simple algebraic group over an algebraically closed field. The paper reduced the odd-type "uniqueness case" (a proper 2-generated core) to a minimal connected simple group with a strongly embedded subgroup and flagged three gaps: removal of the tameness assumption, full elimination of the strongly embedded configuration, and integration into a tameness-free classification. The first two gaps are closed in the published $K^*$-setting: Borovik–Burdges's tameness-free new trichotomy theorem, and Burdges–Cherlin–Jaligot's theorem that a minimal connected simple group of odd type with a proper definable strongly embedded subgroup has Prüfer 2-rank 1, so that generic-type $K^*$-groups are algebraic. The surviving open core lies in the current third-wave $L^*$-theory with the NTA2 condition: eliminate the strongly embedded configuration at Prüfer 2-rank at least 2 (there only "2-rank equals Prüfer rank" is known, versus Prüfer rank 1 in the $K^*$ case), and complete the quasithin classification at Prüfer 2-rank 2 (PSp4 at 2-rank at least 4; G2 versus an exotic characteristic-3 configuration at 2-rank 3; the unexplored 2-rank 2 case), whose claimed proofs exist only in unvetted 2022 drafts. The objective is a theorem that every simple $L^*$-group of finite Morley rank and odd type satisfying NTA2 with Prüfer 2-rank at least 2 is a Chevalley group over an algebraically closed field of characteristic not 2 — or an explicit non-algebraic counterexample.

Background

A group of finite Morley rank is a group definable in a totally transcendental first-order theory; Morley rank is a model-theoretic dimension, and "definable" means first-order definable (with parameters) in the ambient structure. Every algebraic group over an algebraically closed field, equipped with its Zariski structure, has finite Morley rank. The Cherlin–Zilber algebraicity conjecture asserts the converse for simple groups: every infinite simple group of finite Morley rank is a simple algebraic group (Chevalley group) over an algebraically closed field. The conjecture remains one of the central open problems of model-theoretic algebra.

The Borovik program attacks it by transplanting the strategy of the classification of finite simple groups (CFSG). Sylow 2-subgroups of a group of finite Morley rank are conjugate, and the connected component $S^\circ$ of a Sylow 2-subgroup $S$ is a central product $B*T$ of a definable connected nilpotent subgroup $B$ of bounded exponent and a 2-torus $T$, a divisible abelian 2-group that is a direct product of finitely many copies of the Prüfer 2-group $\mathbb{Z}_{2^\infty}$. One then divides by 2-structure: degenerate type (no involutions), odd type (involutions present, $S^\circ$ a nontrivial 2-torus, hence finite 2-rank), and even and mixed type (infinite 2-rank). Odd type is where algebraic groups have base field of characteristic not 2. The 2-rank of a group is the maximal rank of an elementary abelian 2-subgroup, and the normal 2-rank is the maximal rank of a normal elementary abelian 2-subgroup; the Prüfer 2-rank, the 2-rank of a maximal 2-torus, corresponds to the Lie rank, and Prüfer 2-ranks 1, 2, and at least 3 are called thin, quasithin, and generic type. The infinite-2-rank (even and mixed) cases of the conjecture were settled in the monograph of Altınel, Borovik, and Cherlin, so finite 2-rank — and odd type in particular — is the remaining terrain.

A $K^*$-group is one whose proper definable sections are $K$-groups (groups all of whose definable connected simple sections are algebraic); a minimal counterexample to the conjecture is a $K^*$-group. A Borel subgroup is a maximal connected definable solvable subgroup, and a minimal connected simple group is one all of whose proper definable connected subgroups are solvable. For a Sylow 2-subgroup $S$, the 2-generated core $\Gamma_{S,2}(G)$ is the definable hull of the subgroup generated by the normalizers $N_G(V)$ of the elementary abelian 2-subgroups $V\le S$ of rank two; a proper 2-generated core is the analog of CFSG's "uniqueness case", a "black hole" subgroup through which local analysis cannot see all of $G$. A proper definable subgroup $M$ is strongly embedded if it contains involutions but $M\cap M^g$ contains none for $g\notin M$: the terminal, most extreme uniqueness configuration (in finite group theory, Bender's theorem identifies such groups with rank-one Lie type groups). Signalizer functors, also imported from CFSG, are definable assignments $i\mapsto\theta(i)\le O(C_G(i))$, where $i$ is an involution, $C_G(i)$ is its centralizer, and $O(H)$ denotes the largest definable connected normal solvable subgroup of $H$; they are used to analyze components — quasisimple subnormal subgroups — in centralizers of involutions.

Early work (Borovik's 1995 trichotomy theorem; Cherlin–Jaligot 2004) proceeded under a tameness hypothesis: the group involves no bad field, i.e. no field of finite Morley rank with a proper infinite definable multiplicative subgroup. Bad fields are believed to exist in characteristic zero, so tameness had to be eliminated. The source paper, Borovik–Burdges–Nesin (2008), made no use of tameness and proved its Strong Embedding Theorem: in a simple $K^*$-group of odd type with normal 2-rank at least 3 and Prüfer 2-rank at least 2, a proper 2-generated core $M=\Gamma_{S,2}(G)$ forces $G$ to be a minimal connected simple group in which $M$ is strongly embedded, with $M=N_G(B)$ for the Borel subgroup $B=M^\circ$. The companion paper of Burdges, Cherlin, and Jaligot (2007) — the elimination announced in the source paper — then proved that a minimal connected simple group of odd type with a proper definable strongly embedded subgroup has Prüfer 2-rank 1, so that a nonalgebraic simple $K^*$-group of odd type has Prüfer 2-rank at most 2, with no tameness; together with Borovik–Burdges's tameness-free new trichotomy theorem and Burdges's signalizer-funnel work, this yields the published classification of generic-type $K^*$-groups as algebraic.

The third wave of the program, the $L^*$-theory, relativizes the hypotheses to odd type itself: an $L$-group of odd type is one whose definable simple sections of odd type are Chevalley groups, with $L^*$ the corresponding condition on proper sections. Because the orthogonality principle behind the even-type theory (Altınel's lemma) has no odd-type analog — the multiplicative group of a characteristic-zero field acts definably on the additive group — the odd-type $L^*$-theory must be supplemented by the condition NTA2: whenever a definable section acts definably on a degenerate-type simple section, any 2-torus in the acting section acts trivially ("no definable 2-toral actions"). Under $L^*$ plus NTA2 the theory parallels the $K^*$-theory with "solvable" replaced by D-groups (connected groups whose connected simple definable sections are all of degenerate type), and the 2-generated-core-to-strong-embedding bridge survives via the published generation theorem; but most of its detailed development exists only in unpublished 2022 drafts of Burdges and Cherlin.

The current frontier, as documented in Cherlin's 2024 survey "Around the algebraicity problem in odd type": at Prüfer 2-rank at least 3 the classification stops at "Chevalley group or strongly embedded subgroup", and in the strongly embedded case only 2-rank = Prüfer 2-rank is known — versus the $K^*$-theorem's Prüfer rank 1 — so that "the uniqueness branch of the $L^*$-theory remains open in general"; at Prüfer 2-rank 2 the claimed identifications ($\mathrm{PSp}_4$ at 2-rank at least 4; $G_2$ or an exotic characteristic-3 configuration at 2-rank 3) are unpublished and unvetted, and at 2-rank 2, with target $\mathrm{PSL}_3$, even the $K^*$ version is unexplored apart from unpublished early work of Altseimer, because the signalizer functor method fails there. Thin and degenerate types lie beyond the reach of these methods and are not part of this problem.

Problem Statement

This is a derived component of the Cherlin–Zilber algebraicity conjecture, not the conjecture itself: it is the surviving open core of the odd-type branch of the Borovik program as left by Borovik–Burdges–Nesin (2008) and its published successors, namely the completion of the tameness-free classification of simple groups of finite Morley rank of odd type at Prüfer 2-rank at least two, in the third-wave $L^*$-theory framework.

Throughout, let $G$ be a simple $L^*$-group of finite Morley rank and odd type (all of whose proper definable simple sections of odd type are Chevalley groups over algebraically closed fields) satisfying NTA2 (no definable 2-toral action on a degenerate-type simple section), with Prüfer 2-rank at least two. Two linked targets serve one objective.

(1) Uniqueness case / strong embedding — the direct continuation of the source paper. Prove that a proper definable strongly embedded subgroup of such a $G$ forces Prüfer 2-rank one (or cannot occur at all), extending the published $K^*$-theorem of Burdges–Cherlin–Jaligot from minimal connected simple groups to this setting. At 2-rank at least three this configuration is likewise reached from a proper 2-generated core $\Gamma_{S,2}(G)$ via the published generation theorem of Burdges–Cherlin. This closes the strongly embedded alternative in the generic-case classification, so that every simple NTA2 $L^*$-group of odd type with Prüfer 2-rank at least three is a Chevalley group over an algebraically closed field of characteristic not two.

(2) Quasithin case. Classify the simple NTA2 $L^*$-groups of odd type with Prüfer 2-rank exactly two: identify $G$ as $\mathrm{PSp}_4$ over an algebraically closed field when the 2-rank is at least four; settle 2-rank three by proving $G$ is a $G_2$ over an algebraically closed field of characteristic not two, or by eliminating the exotic configuration of condition (*) in Theorem 1.18 of Cherlin's 2024 survey (all involutions conjugate, involution centralizer $\mathrm{SL}_3*\mathrm{SL}_3$ over equal-rank characteristic-three fields, with the stated normalizer condition); and treat 2-rank two, where the expected target is $\mathrm{PSL}_3$. Since the identifications at 2-rank at least three currently exist only in unpublished and unvetted drafts of Burdges–Cherlin, whose proofs are not available in vetted form.

The single objective encompassing both targets: prove that every simple group of finite Morley rank and odd type with Prüfer 2-rank at least two, satisfying $L^*$ and NTA2, is a Chevalley group over an algebraically closed field of characteristic not two — or else construct explicitly a non-algebraic example of such a group. Thin type (Prüfer 2-rank one) and degenerate type lie outside the scope of this problem: there the $L^*$ and signalizer-funnel methods break down and the analysis changes character.

References

1. Alexandre V. Borovik, Jeffrey Burdges, Ali Nesin, "Uniqueness cases in odd-type groups of finite Morley rank", Journal of the London Mathematical Society (2) 77 (2008), no. 1, 240-252. DOI: 10.1112/jlms/jdm106; preprint arXiv:0711.4167, https://arxiv.org/abs/0711.4167
2. Jeffrey Burdges, Gregory Cherlin, Eric Jaligot, "Minimal connected simple groups of finite Morley rank with strongly embedded subgroups", Journal of Algebra 314 (2007), no. 2, 581-612. DOI: 10.1016/j.jalgebra.2006.12.031; preprint arXiv:0711.4165, https://arxiv.org/abs/0711.4165
3. Alexandre Borovik, Jeffrey Burdges, "A new trichotomy theorem for groups of finite Morley rank", Journal of the London Mathematical Society (2) 77 (2008), no. 1, 1-14. DOI: 10.1112/jlms/jdm088; preprint arXiv:0711.4169, https://arxiv.org/abs/0711.4169
4. Jeffrey Burdges, "Signalizers and balance in groups of finite Morley rank", Journal of Algebra 321 (2009), no. 5, 1383-1406. DOI: 10.1016/j.jalgebra.2008.12.001; preprint arXiv:0711.4210, https://arxiv.org/abs/0711.4210
5. Jeffrey Burdges, Gregory Cherlin, "A generation theorem for groups of finite Morley rank", Journal of Mathematical Logic 8 (2008), no. 2, 163-195. DOI: 10.1142/s0219061308000750; preprint arXiv:0801.3957, https://arxiv.org/abs/0801.3957
6. Gregory Cherlin, Eric Jaligot, "Tame minimal simple groups of finite Morley rank", Journal of Algebra 276 (2004), no. 1, 13-79. DOI: 10.1016/j.jalgebra.2003.12.027, https://doi.org/10.1016/j.jalgebra.2003.12.027
7. A. V. Borovik, "Simple locally finite groups of finite Morley rank and odd type", in: Finite and locally finite groups (Istanbul, 1994), NATO ASI Series C: Mathematical and Physical Sciences 471, Kluwer Academic Publishers, Dordrecht, 1995, pp. 247-284. ISBN 0-7923-3669-0; Zbl 0840.20021, https://zbmath.org/?q=an%3A0840.20021
8. Tuna Altınel, Alexandre V. Borovik, Gregory Cherlin, "Simple Groups of Finite Morley Rank", Mathematical Surveys and Monographs 145, American Mathematical Society, Providence, RI, 2008. DOI: 10.1090/surv/145, ISBN 978-0-8218-4305-5
9. Gregory Cherlin, "Around the algebraicity problem in odd type", Model Theory 3 (2024), no. 2, 505-538. DOI: 10.2140/mt.2024.3.505, https://msp.org/mt/2024/3-2/p13.xhtml
10. Jeffrey Burdges, Gregory Cherlin, "L*-Groups of odd type with restricted 2-toral automorphisms, I: High Prüfer 2-rank; II: Prüfer 2-rank 2 and 2-rank at least 3; III: Identification of PSp4; IV: Toward the identification of G2", unpublished drafts (2022), https://sites.math.rutgers.edu/~cherlin/Paper/inprep.html
11. Alexandre V. Borovik, Jeffrey Burdges, Ali Nesin, "Uniqueness Cases in Odd Type Groups of Finite Morley Rank, Revisited", preprint (2011), arXiv:1111.7187, https://arxiv.org/abs/1111.7187 — unrefereed preprint; its Strong Embedding and Uniqueness Subgroup Theorems are stated only for Prüfer 2-rank at least three and under a $K$-group hypothesis on the proper definable subgroups containing a maximal 2-torus, so they do not cover the $L^*$/NTA2 target (1).
12. Ayşe Berkman, 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 — unrefereed preprint; its identification is framed for the Hrushovski Programme under a generic-automorphism hypothesis and its programme is explicitly incomplete, so it does not settle the generic branch of target (1).