Standardness of sharply 2-transitive groups of finite Morley rank (Borovik–Nesin conjecture)

A permutation group is sharply 2-transitive if every ordered pair of distinct points is carried to every other such pair by exactly one group element. The only known infinite examples of finite Morley rank — the model-theoretic analogue of algebraic dimension — are the one-dimensional affine groups $\mathrm{AGL}_1(K)=K^+\rtimes K^\times$ of algebraically closed fields, called standard, and Borovik and Nesin conjectured in 1994 that these are the only ones. The conjecture separates into a splitting problem (does the group contain a regular abelian normal subgroup, equivalently is its associated near-domain a near-field?) and a recognition problem (is a split group the affine group of an algebraically closed field?). Subsequent work has settled recognition in characteristic $\neq 2$, splitting in characteristics $2$ and $3$, splitting at Morley rank $6$ and for nilpotent point stabilizers, and hence the full conjecture in characteristic $3$; as recorded in Tent's 2026 ICM survey, the surviving open core is precisely: (1) does every sharply 2-transitive group of finite Morley rank of permutation characteristic $0$ or odd prime $p\ge 5$ split, and (2) is every near-field of finite Morley rank and characteristic $2$ an algebraically closed field? The stakes exceed the class itself: a non-split example of finite Morley rank with strongly minimal translation centralizers would be a simple group, hence a counterexample to the Cherlin–Zilber algebraicity conjecture.

Background

Morley rank is the model-theoretic analogue of algebraic dimension: an ordinal-valued rank on the definable sets of a first-order structure, defined by an inductive dimension calculus, which over algebraically closed fields coincides with Zariski dimension. A structure has finite Morley rank if all of its definable sets have finite Morley rank; such structures are in particular $\omega$-stable (over any countable parameter set there are only countably many complete types). Groups definable in such structures, called groups of finite Morley rank, imitate algebraic groups over algebraically closed fields in many respects: a classical theorem of the subject (Macintyre) says that any field of finite Morley rank is algebraically closed, and Zilber's indecomposability theorem — in a group of finite Morley rank, the subgroup generated by an arbitrary family of definable subsets is itself definable and is already a finite product of them — supplies a substitute for the algebraic-group toolkit (it is treated in the Borovik–Nesin monograph). The central open problem of the area is the Cherlin–Zilber algebraicity conjecture, asserting that every simple group of finite Morley rank is an algebraic group over an algebraically closed field; it is proved in even type (when the group contains an infinite elementary abelian $2$-subgroup) and remains open both in odd type (Sylow $2$-subgroups contain a nontrivial divisible abelian subgroup) and in the degenerate case of groups without involutions, where no analogue of character theory exists.

A permutation group $G$ on a set $X$ is sharply 2-transitive if for any two ordered pairs $(x_1,x_2)$ and $(y_1,y_2)$ of distinct points there is exactly one element $g\in G$ with $g(x_1)=y_1$ and $g(x_2)=y_2$. The point stabilizer $B=G_x$ is then a malnormal subgroup ($B\cap B^g=\{1\}$ for $g\notin B$), so $G$ is a Frobenius group. The standard examples are the one-dimensional affine groups $\mathrm{AGL}_1(K)=K^+\rtimes K^\times$ of transformations $x\mapsto ax+b$ with $a\neq 0$ over a field $K$; a sharply 2-transitive group is called standard if it is of this form for an algebraically closed field $K$. Such a group splits if it has a regular (i.e. sharply 1-transitive) normal subgroup $N$; then $G=N\rtimes B$, the subgroup $N$ is automatically abelian, and $B$ acts by conjugation as a group of automorphisms of $N$ acting regularly on $N\setminus\{1\}$. A classical correspondence associates to every sharply 2-transitive group the group of affine transformations $x\mapsto a+b\cdot x$ of a near-domain: a set $K$ with two operations such that $(K,+)$ is a loop (the equations $a+x=b$ and $y+a=b$ have unique solutions), the nonzero elements form a group under multiplication, left distributivity $a\cdot(b+c)=a\cdot b+a\cdot c$ holds, and elements $d_{a,b}$ measure the failure of associativity of addition via $a+(b+x)=(a+b)+d_{a,b}\cdot x$. The near-domain is a near-field exactly when addition is associative ($d_{a,b}=1$), and this happens exactly when the group splits; a near-field is a skew field (division ring) precisely when right distributivity also holds. Thus splitting and recognition are respectively the questions 'is the near-domain a near-field?' and 'is the near-field a field?'.

All involutions of a sharply 2-transitive group are conjugate, which yields a notion of characteristic: permutation characteristic $2$ if involutions have no fixed points; otherwise the products of two distinct involutions form a single conjugacy class of elements of common order $p$ (a prime) or of infinite order, and the characteristic is $p$ or $0$. Finite sharply 2-transitive groups were classified in the 1930s: they all split and correspond to finite near-fields, which need not be commutative. For infinite groups the class was believed rigid until Rips–Segev–Tent constructed the first non-split sharply 2-transitive groups (characteristic $2$, into which any group embeds); Tent–Ziegler gave a simpler construction whose example embeds in a virtually free, hence linear, group, Rips–Tent produced non-split examples of characteristic $0$, and Amelio–André–Tent built non-split examples for every sufficiently large odd prime characteristic. None of these examples has finite Morley rank: by Zilber's indecomposability theorem, in a sharply 2-transitive group of finite Morley rank there is a uniform bound on the number of involutions needed to express any element, and the known constructions violate this and related criteria. Positive splitting results without finite Morley rank include characteristic $3$ (Kerby–Wefelscheid, with a short proof by Türkelli), local compactness in the connected case, local finiteness, linearity in characteristic $\neq 2$, and definability in an o-minimal structure.

Within finite Morley rank, Borovik and Nesin conjectured in their 1994 monograph that every infinite sharply 2-transitive permutation group of finite Morley rank is standard; they conjectured more generally that Frobenius groups of finite Morley rank split with nilpotent kernel. The sharply 3-transitive analogue was settled long ago — every sharply 3-transitive group of finite Morley rank is $\mathrm{PSL}_2(K)$ over an algebraically closed field (Nesin) — which is why the sharply 2-transitive case, a rank-1 phenomenon, is the stubborn one. The known cases of the conjecture are: characteristic $3$ completely (classical splitting by Kerby–Wefelscheid plus recognition in characteristic $\neq 2$ proved in [1], whose characteristic-$0$ part goes back to Cherlin–Grundhöfer–Nesin–Völklein in 1991); splitting in characteristic $2$ (Altınel–Berkman–Wagner, using a 2-structure theorem also obtained by Deloro–Wiscons); recognition of all split groups in characteristic $\neq 2$; splitting for nilpotent point stabilizers (Borovik–Nesin); and splitting at Morley rank $6$ (Clausen–Tent). What remains is exactly two questions, recorded as such in Tent's 2026 ICM survey: splitting in permutation characteristic $0$ and odd primes $p\ge 5$, and whether near-fields of finite Morley rank of characteristic $2$ are algebraically closed fields. The problem matters well beyond its own class: sharply 2-transitive groups are the standardly cited potential source of counterexamples to the Cherlin–Zilber conjecture, and Clausen–Tent showed that a non-split sharply 2-transitive group of finite Morley rank whose centralizers of translations are strongly minimal (of Morley rank and degree $1$) would be simple — an outright algebraicity counterexample.

Problem Statement

Let $G$ be an infinite sharply 2-transitive permutation group of finite Morley rank, i.e. a group definable in a structure of finite Morley rank acting definably and sharply 2-transitively on a definable set. The Borovik–Nesin conjecture asserts that $G$ is standard: isomorphic to $\mathrm{AGL}_1(K)=K^+\rtimes K^\times$ for an algebraically closed field $K$ (necessarily of the same permutation characteristic), acting on the affine line of $K$.

In the present state of knowledge the conjecture is equivalent to the conjunction of the following two open statements:

(1) Splitting in characteristic $0$ and $p\ge 5$: if the permutation characteristic of $G$ is $0$ or an odd prime $p\ge 5$, then $G$ splits, i.e. contains a regular abelian normal subgroup; equivalently, the near-domain associated to $G$ is a near-field. (Already known, hence not part of what must be proved: characteristic $2$; characteristic $3$; nilpotent point stabilizer; Morley rank $6$.)

(2) Recognition in characteristic $2$: every near-field of finite Morley rank and characteristic $2$ is an algebraically closed field; equivalently — since sharply 2-transitive groups of finite Morley rank and characteristic $2$ are now known to split — every sharply 2-transitive group of finite Morley rank and permutation characteristic $2$ is isomorphic to $\mathrm{AGL}_1(K)$ for an algebraically closed field $K$ of characteristic $2$. (Already known: all characteristics $\neq 2$, and the case of soluble point stabilizers.)

Resolve the Borovik–Nesin conjecture in the stated generality — with no restriction on rank, characteristic, or stabilizer beyond those explicitly listed in (1) and (2). A non-standard infinite sharply 2-transitive group of finite Morley rank — non-split, or split with a near-field that is not an algebraically closed field — would settle it negatively.

References

1. Tuna Altinel, Ayse Berkman, Frank Olaf Wagner, Sharply 2-transitive groups of finite Morley rank, arXiv:1811.10854 (v1 27 Nov 2018; v2 19 Mar 2024; v3 4 Jun 2026), DOI 10.48550/arXiv.1811.10854, https://arxiv.org/abs/1811.10854.
2. Katrin Tent, From the Cherlin–Zilber Conjecture via Sharply 2-transitive Groups to the Burnside Problem, Proceedings of the International Congress of Mathematicians 2026, Volume 3: Invited Lectures (Sections 1–4), SIAM, 2026, pp. 113–129, DOI 10.1137/25m1805266, arXiv:2606.18207, https://arxiv.org/abs/2606.18207.
3. Alexandre Borovik, Ali Nesin, Groups of Finite Morley Rank, Oxford Logic Guides 26, The Clarendon Press, Oxford University Press, New York, 1994, DOI 10.1093/oso/9780198534457.001.0001, https://doi.org/10.1093/oso/9780198534457.001.0001.
4. Tim Clausen, Katrin Tent, Mock hyperbolic reflection spaces and Frobenius groups of finite Morley rank, Model Theory 2 (2023), no. 2, 137–175, DOI 10.2140/mt.2023.2.137, https://doi.org/10.2140/mt.2023.2.137.
5. Gregory Cherlin, Theo Grundhöfer, Ali Nesin, Helmut Völklein, Sharply transitive linear groups over algebraically closed fields, Proceedings of the American Mathematical Society 111 (1991), no. 2, 541–550, DOI 10.1090/s0002-9939-1991-1031450-5, https://doi.org/10.1090/s0002-9939-1991-1031450-5.
6. W. Kerby, H. Wefelscheid, Bemerkungen über Fastbereiche und scharf zweifach transitive Gruppen, Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg 37 (1972), 20–29, DOI 10.1007/BF02993896, https://doi.org/10.1007/BF02993896.
7. William Kerby, On infinite sharply multiply transitive groups, Hamburger Mathematische Einzelschriften, Neue Folge, Heft 6, Vandenhoeck & Ruprecht, Göttingen, 1974, zbMATH 3455513 (Zbl 0291.20039), https://zbmath.org/3455513.
8. Seyfi Türkelli, Splitting of sharply 2-transitive groups of characteristic 3, Turkish Journal of Mathematics 28 (2004), no. 3, 295–298, zbMATH 2166481, arXiv:0809.0959, https://arxiv.org/abs/0809.0959.
9. Eliyahu Rips, Yoav Segev, Katrin Tent, A sharply 2-transitive group without a non-trivial abelian normal subgroup, Journal of the European Mathematical Society 19 (2017), no. 10, 2895–2910, DOI 10.4171/jems/730, https://doi.org/10.4171/jems/730.
10. Eliyahu Rips, Katrin Tent, Sharply 2-transitive groups of characteristic 0, Journal für die reine und angewandte Mathematik 750 (2019), 227–238, DOI 10.1515/crelle-2016-0054, https://doi.org/10.1515/crelle-2016-0054.
11. Marco Amelio, Simon André, Katrin Tent, Non-split Sharply 2-Transitive Groups of Odd Positive Characteristic, International Mathematics Research Notices 2025, no. 19, article rnaf294, DOI 10.1093/imrn/rnaf294, https://doi.org/10.1093/imrn/rnaf294.
12. Katrin Tent, Martin Ziegler, Sharply 2-transitive groups, Advances in Geometry 16 (2016), no. 1, 131–134, DOI 10.1515/advgeom-2015-0047, https://doi.org/10.1515/advgeom-2015-0047.
13. Adrien Deloro, 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, https://doi.org/10.1112/blms.12334.
14. Frank Wagner, Some remarks on sharply 2-transitive groups and near-domains, arXiv:2202.13740 (2022), https://arxiv.org/abs/2202.13740.
15. Tuna Altınel, Alexandre 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, https://doi.org/10.1090/surv/145.
16. Samuel Zamour, Symétrons et K-boucles $\omega$-stables, Archive for Mathematical Logic 64 (2025), 857–876, DOI 10.1007/s00153-025-00967-4, arXiv:2309.10611, https://arxiv.org/abs/2309.10611.
17. J. de la Nuez González, A simple acylindrical recipe for non-split characteristic 2 sharply k-transitive actions and their generalizations, arXiv:2608.10114 (2026), https://arxiv.org/abs/2608.10114.
18. Ali Nesin, On sharply n-transitive superstable groups, Journal of Pure and Applied Algebra 69 (1990), no. 1, 73–88, DOI 10.1016/0022-4049(90)90079-W, https://doi.org/10.1016/0022-4049(90)90079-W.
19. Katrin Tent, Sharply 3-transitive groups, Advances in Mathematics 286 (2016), 722–728, DOI 10.1016/j.aim.2015.09.018, https://doi.org/10.1016/j.aim.2015.09.018.
20. V. D. Mazurov, E. I. Khukhro (eds.), The Kourovka Notebook. Unsolved Problems in Group Theory, arXiv:1401.0300, https://arxiv.org/abs/1401.0300 (Question 12.48).