In their 2008 preprint on definably linear groups of finite Morley rank, Alexandre Borovik and Jeffrey Burdges proved that an infinite simple definable subgroup of GLn(k)\mathrm{GL}_n(k) over a field kk of finite Morley rank and characteristic zero, not Zariski closed, contains no involutions, and they closed the preprint with a conjecture designed to extend the same method: there is no simple group of finite Morley rank in which every strongly real element — every element inverted by an involution — lies inside some C∘(T)C^\circ(T), the definable connected component of the centralizer of a Sylow∘^\circ 2-subgroup TT; as throughout this literature, the quantifier is over infinite simple groups. The conjecture measures how far Bachmann-style involution geometry can reach in eliminating the odd-type configurations left open by the Cherlin–Zilber algebraicity conjecture. Deloro and Wiscons (2020) settled the trivial-intersection case, generalized the Borovik–Burdges reduction, and re-proved its main theorem, but state explicitly that they could not prove the conjecture itself, which they restate for Cartan subgroups; the conjecture remains open.

Background

A group of finite Morley rank is a group equipped with a finite-valued dimension function, the rank, on the family of its first-order definable subsets, satisfying the same combinatorial axioms as Zariski dimension in algebraic geometry; the class was introduced by model theorists, and it contains the groups of KK-points of every algebraic group over an algebraically closed field KK, with rank equal to algebraic dimension. “Definable” means definable in the ambient structure, and for a definable subgroup HH the notation H∘H^\circ denotes its connected component, the smallest definable subgroup of finite index, so that C∘(T)C^\circ(T) below means (CG(T))∘(C_G(T))^\circ. The central open problem of the area is 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. The conjecture is open, and the surviving burden is organized by the 2-structure, in close analogy with the classification of finite simple groups: an involution is an element of order two, and by the Even and Mixed Type Theorem of Altınel, Borovik and Cherlin a simple group of finite Morley rank containing an infinite elementary abelian 2-subgroup is a Chevalley group (the group of KK-points of a simple algebraic group) over an algebraically closed field of characteristic two, so the open cases are odd type (involutions present, but no infinite elementary abelian 2-subgroup) and degenerate type (no involutions at all, equivalently finite Sylow 2-subgroups, by a theorem of Borovik, Burdges and Cherlin); whether infinite simple groups of degenerate type exist is itself open. Borovik and Burdges refer to the program of classifying simple groups of finite Morley rank containing an involution as the “L∗L^*” project, and their preprint is a contribution to it.

An element of a group is strongly real if it is inverted by some involution, equivalently if it is a product of two involutions. Since Brauer's school, the distribution of strongly real elements and of centralizers of involutions is a standard handle on group structure, and the same is true in the model-theoretic setting. A Sylow 2-subgroup of a group of finite Morley rank is a maximal 2-subgroup; Sylow 2-subgroups are conjugate (Borovik–Nesin, Theorem 10.11), and a Sylow∘^\circ 2-subgroup is the connected component S∘S^\circ of such a subgroup SS. In groups without 2-unipotent torsion — the relevant setting here — a Sylow∘^\circ 2-subgroup is just a maximal 2-torus, that is, a definable divisible abelian 2-group, necessarily a finite product of copies of the Prüfer quasi-cyclic group Z/2∞Z\mathbb{Z}/2^\infty\mathbb{Z}. A Borel subgroup is a maximal definable connected solvable subgroup, and a Cartan subgroup is the centralizer of a maximal decent torus (a divisible abelian torsion subgroup with good conjugacy behavior).

Borovik and Burdges proved that an infinite simple definable subgroup GG of GLn(k)\mathrm{GL}_n(k), where kk is a field of finite Morley rank and characteristic zero and GG is not Zariski closed, has no involutions and has self-normalizing Borel subgroups; Poizat had previously settled the analogous characteristic-pp statement, and Borel-conjugacy input in definably linear groups comes from Mustafin. The geometric core of their proof runs in a connected group of finite Morley rank without unipotent torsion in which Borel subgroups are nilpotent and hereditarily conjugate (every definable connected subgroup enjoys conjugacy of its own Borel subgroups): there, the centralizers of distinct Sylow∘^\circ 2-subgroups are disjoint, every strongly real element lies in the union of the conjugates of C∘(T)C^\circ(T), and the involutions carry the incidence structure of a projective plane in which three involutions are collinear exactly when their product is an involution. Bachmann's classical theorem on reflection geometries then identifies the group generated by the involutions with SO3(k,f)\mathrm{SO}_3(k,f) for an interpretable field kk and non-isotropic quadratic form ff, which is incompatible with finite Morley rank, so such a simple GG cannot have involutions at all. The problem recorded here is the conjecture with which the preprint concludes: the same covering configuration should be impossible in a simple group of finite Morley rank with no auxiliary hypotheses at all. Deloro and Wiscons (2020) built the modern theory of exactly this circle of ideas: their Geometric Theorem proves the trivial-intersection case — in a connected group of finite Morley rank with involutions and no infinite elementary abelian 2-subgroup, the conjugates of a definable connected subgroup that is TI (intersecting each distinct conjugate trivially) and almost self-normalizing (of finite index in its normalizer) can never cover all strongly real elements — and their Theorem B generalizes the Borovik–Burdges reduction while re-proving its main theorem without Bachmann's theorem. They identify the obstruction that stops the method: unlike in the algebraic world, Borel subgroups need not be conjugate inside definable subgroups, so the disjoint-centralizer configuration cannot be forced; the surviving core of the conjecture is therefore the non-TI case, in which the connected centralizers of distinct Sylow∘^\circ 2-subgroups intersect nontrivially, together with the adjacent configurations currently under study in the odd-type program: strongly embedded subgroups (proper subgroups of even order meeting each distinct conjugate in odd order) and quasi-Frobenius pairs (a connected group of finite Morley rank with a definable subgroup that is TI and of finite index in its normalizer, generalizing the point stabilizers of a Frobenius group).

Problem Statement

Prove or refute the following conjecture of Borovik and Burdges (the closing Conjecture 1 of arXiv:0801.3958), stated here in its original, source-faithful form.

There is no simple group of finite Morley rank in which every strongly real element lies inside some C∘(T)C^\circ(T) for TT a Sylow∘^\circ 2-subgroup.

Here an element is strongly real if it is inverted by some involution (equivalently, is a product of two involutions); TT ranges over Sylow∘^\circ 2-subgroups, i.e. connected components S∘S^\circ of Sylow 2-subgroups SS (maximal 2-subgroups); C∘(T)C^\circ(T) is the definable connected component (CG(T))∘(C_G(T))^\circ of its centralizer; and “every strongly real element lies inside some C∘(T)C^\circ(T)” means that for each strongly real element xx of GG there is such a TT, possibly depending on xx, with x∈C∘(T)x\in C^\circ(T). Since Sylow 2-subgroups of a group of finite Morley rank are conjugate (Borovik–Nesin, Theorem 10.11), the condition is equivalent to: the set of strongly real elements is contained in ⋃g∈GC∘(T)g\bigcup_{g\in G} C^\circ(T)^g for a single fixed Sylow∘^\circ 2-subgroup TT.

Two conventions fix the scope. First, the source's Conjecture 1 is written for a “simple group of finite Morley rank” with no infinitude qualifier, and the phrase is understood here, by the standing convention of the field (the Cherlin–Zilber algebraicity conjecture itself is stated for infinite simple groups), to mean an infinite simple group: every finite group has finite Morley rank, and a finite simple group of odd order (for example Z/3\mathbb{Z}/3) has no involutions and hence no strongly real elements, so the covering condition holds vacuously for it; without this convention, such groups would trivially refute the literal statement, and no such trivial refutation is intended. (For a finite simple group with involutions the computation is different: T=S∘=1T=S^\circ=1, CG(T)=GC_G(T)=G, and G∘=1G^\circ=1 by the definition of H∘H^\circ above, so C∘(T)=1C^\circ(T)=1 does not contain the involutions; the finite setting is in any case not the intended model-theoretic context.) Second, the substantive case, the one intended by the source's proof context and by the Deloro–Wiscons restatement — “there is no simple ranked group in which all strongly real elements lie in ⋃GCg\bigcup_G C^g for CC a Cartan subgroup” — is that of simple groups containing involutions, where the Sylow∘^\circ 2-subgroups are nontrivial 2-tori and the conjecture asserts that the conjugates of their connected centralizers can never cover all strongly real elements.

Read literally, the statement also excludes involution-free simple groups, for which the covering condition holds vacuously, and thereby subsumes the open problem of whether simple groups of finite Morley rank of degenerate type exist.

References

  1. Alexandre Borovik and Jeffrey Burdges, “Linear groups of finite Morley rank” (preprint; the paper's own heading is “Definably linear groups of finite Morley rank”; never journal-published — the identically titled paper by Borovik alone in J. Inst. Math. Jussieu 7 (2008) is a separate Russian-language survey), arXiv:0801.3958, 2008. https://arxiv.org/abs/0801.3958
  2. Adrien Deloro and Joshua Wiscons, “The geometry of involutions in ranked groups with a TI-subgroup”, Bulletin of the London Mathematical Society 52 (2020), no. 3, 411–428, DOI 10.1112/blms.12334 (preprint arXiv:1901.04453). https://doi.org/10.1112/blms.12334
  3. Tuna Altınel, Alexandre Borovik, and Gregory Cherlin, “Simple Groups of Finite Morley Rank”, Mathematical Surveys and Monographs 145, American Mathematical Society, Providence, RI, 2008, ISBN 978-0-8218-4305-5, DOI 10.1090/surv/145. https://doi.org/10.1090/surv/145
  4. Alexandre Borovik, Jeffrey Burdges, and Gregory Cherlin, “Involutions in groups of finite Morley rank of degenerate type”, Selecta Mathematica (New Series) 13 (2007), no. 1, 1–22, DOI 10.1007/s00029-007-0030-z. https://doi.org/10.1007/s00029-007-0030-z
  5. Alexandre Borovik and Ali Nesin, “Groups of Finite Morley Rank”, Oxford Logic Guides 26, The Clarendon Press, Oxford University Press, New York, 1994, ISBN 0-19-853445-0, DOI 10.1093/oso/9780198534457.001.0001, Zbl 0816.20001. https://doi.org/10.1093/oso/9780198534457.001.0001
  6. Friedrich Bachmann, “Aufbau der Geometrie aus dem Spiegelungsbegriff”, Die Grundlehren der mathematischen Wissenschaften 96, Springer-Verlag, Berlin, 1959, DOI 10.1007/978-3-662-01234-5, ISBN 978-3-662-01235-2. https://doi.org/10.1007/978-3-662-01234-5
  7. Bruno Poizat, “Quelques modestes remarques à propos d'une conséquence inattendue d'un résultat surprenant de Monsieur Frank Olaf Wagner”, The Journal of Symbolic Logic 66 (2001), no. 4, 1637–1646, DOI 10.2307/2694966. https://doi.org/10.2307/2694966
  8. Yerulan Mustafin, “Structure des groupes linéaires définissables dans un corps de rang de Morley fini”, Journal of Algebra 281 (2004), no. 2, 753–773, DOI 10.1016/j.jalgebra.2004.07.017. https://doi.org/10.1016/j.jalgebra.2004.07.017
  9. Adrien Deloro and Éric Jaligot, “Involutive automorphisms of N∘∘N^\circ_\circ-groups of finite Morley rank”, Pacific Journal of Mathematics 285 (2016), no. 1, 111–184, DOI 10.2140/pjm.2016.285.111. https://doi.org/10.2140/pjm.2016.285.111
  10. Tuna Altınel, Luis Jaime Corredor, and Adrien Deloro, “On quasi-Frobenius pairs of finite Morley rank”, Proceedings of the American Mathematical Society 152 (2024), no. 7, 2791–2804, DOI 10.1090/proc/16771 (preprint arXiv:2305.00830). https://doi.org/10.1090/proc/16771
  11. Ayşe Berkman and Alexandre V. Borovik, “The Şiş Kebap Theorem: A Generic Identification Theorem for Groups of Finite Morley Rank”, preprint, arXiv:2608.01161, 2026. https://arxiv.org/abs/2608.01161
  12. Katrin Tent, “From the Cherlin–Zilber Conjecture via sharply 22-transitive groups to the Burnside problem”, preprint, arXiv:2606.18207, 2026. https://arxiv.org/abs/2606.18207