Problem #14

Cherlin-Zilber Algebraicity Conjecture

Open!!

Every infinite simple group of finite Morley rank is an algebraic group over an algebraically closed field.

Model Theory

Known Partial Results

It is known for infinite simple groups of Morley rank at most three, for groups of 'even type', and for simple groups of finite Morley rank with a split BN-pair of Tits rank at least 2.

Notes

The original sources state the conjecture in different terms: for Zilber [Zil77], the conjecture was for uncountably categorical groups, and for Cherlin [Che79], the conjecture was for infinite simple omega-stable groups. The contemporary formulation splits the difference.

Reference for the problem statement

[Che79]Gregory Cherlin, Groups of small Morley rank, Annals of Mathematical Logic, 1979 [doi]

Additional References

[ABC97]T. Altınel, A. Borovik and G. Cherlin, Groups of mixed type, J. Algebra, 1997

[ABC08]T. Altınel, A. V. Borovik and G. Cherlin, Simple Groups of Finite Morley Rank, Mathematical Surveys and Monographs 145, AMS, 2008

[BN94]A. Borovik and A. Nesin, Groups of Finite Morley Rank, Oxford Logic Guides 26, Oxford University Press, 1994

[Fre18]O. Frécon, Simple groups of Morley rank 3 are algebraic, J. Amer. Math. Soc., 2018

[KTV99]L. Kramer, K. Tent and H. Van Maldeghem, Simple groups of finite Morley rank and Tits buildings, Israel J. Math., 1999

[Nes89]A. Nesin, Nonsolvable groups of Morley rank 3, J. Algebra, 1989

[Nes91]A. Nesin, On bad groups, bad fields, and pseudoplanes, J. Symbolic Logic, 1991

[Ten25]K. Tent, From the Cherlin–Zilber Conjecture via sharply 2-transitive groups to the Burnside problem, survey, 2025

[Zil77]B. I. Zil'ber, Groups and rings whose theory is categorical, Fundamenta Mathematicae, 1977

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