Cherlin–Zilber Algebraicity Conjecture
Canonical statement
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Every infinite simple group \(G\) of finite Morley rank is, as an abstract group, isomorphic to the group of \(K\)-rational points of a simple algebraic group over an algebraically closed field \(K\).Notes
The Cherlin–Zilber algebraicity conjecture asserts that every infinite simple group of finite Morley rank is isomorphic, as an abstract group, to the group of -rational points of a simple algebraic group over an algebraically closed field . Morley rank is a model-theoretic notion of dimension, so the conjecture would identify the abstract simple groups of finite-dimensional model theory with the simple algebraic groups. It goes back to about 1979: Cherlin's study of groups of small Morley rank is an early source [Cherlin1979Morley], and the jointly named modern formulation took shape afterwards.
The conjecture is known in low Morley rank, where the analysis originates with Cherlin [Cherlin1979Morley], and it has been settled in the rank- case; a systematic theory of groups of finite Morley rank is developed by Borovik and Nesin [BorovikNesin1994]. The program also connects to sharply -transitive groups and the Burnside problem [Tent2026CherlinZilber].
Despite these advances, the classification of arbitrary infinite simple groups of finite Morley rank is incomplete, and the full conjecture remains open.
References (3)
- [Cherlin1979Morley]
Groups of small Morley rank
Open ↗Gregory Cherlin · 1979 · misc
- [BorovikNesin1994]
Groups of Finite Morley Rank
Alexandre Borovik and Ali Nesin · 1994 · misc
- [Tent2026CherlinZilber]
From the Cherlin–Zilber conjecture via sharply -transitive groups to the Burnside problem
Open ↗Katrin Tent · 2026 · misc
The boxed statement is the canonical open formulation — not a stronger variant or a related research program. The status reflects the catalog's last review; do your own literature search before investing serious effort.