## Groups of mixed type.(English)Zbl 0878.20003

The paper under review deals with groups of finite Morley rank. The current state of the structure theory of groups of finite Morley rank is presented in the book by A. Borovik, A. Nesin [Groups of finite Morley rank, Clarendon Press, Oxford (1994; Zbl 0816.20001)]. The main stimulus in the field is the open Cherlin-Zil’ber Conjecture which proposes that infinite simple groups of finite Morley rank are algebraic groups over algebraically closed fields. The authors present a program of investigation of groups of finite Morley rank which gives an approach to this problem. The approach is inspired by the ideas from finite group theory developed in the classification of finite simple groups.
The main result of the paper is a proof of a special case of the following conjecture (which is a part of the program and which should be true if the Cherlin-Zil’ber Conjecture is true): a simple group of finite Morley rank cannot be of mixed type. Here a group of finite Morley rank is said to be of mixed type if it is neither of odd type nor of even type; it is called of odd type if its Sylow subgroups are of bounded exponent; it is called of even type if the connected component of any of its Sylow subgroups is divisible abelian. Specifically, the authors prove the conjecture for the so called tame $$K^*$$-groups; these groups naturally arise when one considers minimal counterexamples to certain versions of the Cherlin-Zil’ber Conjecture.

### MSC:

 20A15 Applications of logic to group theory 03C60 Model-theoretic algebra 20E32 Simple groups 03C45 Classification theory, stability, and related concepts in model theory 20G15 Linear algebraic groups over arbitrary fields 20D05 Finite simple groups and their classification 20E34 General structure theorems for groups

Zbl 0816.20001
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### References:

 [1] Altınel, T., Groups of finite Morley rank with strongly embedded subgroups, J. algebra, 180, 778-807, (1996) · Zbl 0855.20029 [2] T. Altınel, G. Cherlin, On central extensions of algebraic groups · Zbl 0932.03042 [3] T. Altınel, G. Cherlin, L.-J. Corredor, A. Nesin, A Hall theorem for ω-stable groups · Zbl 0922.20042 [4] O. V. Belegradek, On groups of finite Morley rank, Abstracts of the Eighth International Congress of Logic, Methodology and Philosophy of Science, Moscow, USSR, 17-22 August 1987, LMPS ’87, 100, 102 [5] Birkhoff, G., Lattice theory, (1967), Am. Math. Soc Providence · Zbl 0126.03801 [6] A. Borovik, Simple locally finite groups of finite Morley rank and odd type, Proceedings of NATO ASI on Finite and Locally Finite Groups, Istanbul, Turkey, 1994, 247, 284 [7] Borovik, A.; Nesin, A., On the schur – zassenhaus theorem for groups of finite Morley rank, J. symbolic logic, 57, 1469-1477, (1992) · Zbl 0773.03025 [8] Borovik, A.; Nesin, A., Schur – zassenhaus theorem revisited, J. symbolic logic, 59, 283-291, (1994) · Zbl 0799.03038 [9] Borovik, A.V.; Nesin, A., Groups of finite Morley rank, (1994), Oxford Univ. Press London · Zbl 0816.20001 [10] A. V. Borovik, B. Poizat, 1990, Simple groups of finite Morley rank without nonnilpotent connected subgroups · Zbl 0731.20019 [11] Borovik, A.V.; Poizat, B., Tores etp, J. symbolic logic, 55, 565-583, (1990) [12] Cherlin, G., Groups of small Morley rank, Ann. math. logic, 17, 1-28, (1979) · Zbl 0427.20001 [13] Corredor, L.-J., Bad groups of finite Morley rank, J. symbolic logic, 54, 768-773, (1989) · Zbl 0689.03017 [14] Hirsch, K., On infinite solvable groups II, Proc. London math. soc. (2), 44, 336-344, (1938) · JFM 64.0066.01 [15] Humphreys, J.E., Introduction to Lie algebras and representation theory, (1980), Springer-Verlag Berlin/New York · Zbl 0254.17004 [16] Humphreys, J.E., Linear algebraic groups, (1981), Springer-Verlag Berlin/New York · Zbl 0507.20017 [17] Macintyre, A., On ω_{1}, Fund. math., 70, 253-270, (1971) [18] Nesin, A., Nonsolvable groups of Morley rank 3, J. algebra, 124, 199-218, (1989) · Zbl 0684.03014 [19] Nesin, A., On sharplyn, J. pure appl. algebra, 69, 73-88, (1990) · Zbl 0732.03029 [20] Nesin, A., On solvable groups of finite Morley rank, Trans. amer. math. soc., 321, 659-690, (1990) · Zbl 0725.03020 [21] Nesin, A., Generalized Fitting subgroup of a group of finite Morley rank, J. symbolic logic, 56, 1391-1399, (1991) · Zbl 0743.03027 [22] Nesin, A., Poly-separated and ω-stable nilpotent groups, J. symbolic logic, 56, 694-699, (1991) · Zbl 0747.03014 [23] Pillay, A.; Sokolovic, Z., A remark on differential algebraic groups, Comm. algebra, 20, 3015-3026, (1992) · Zbl 0766.12004 [24] Poizat, B., Groupes stables, (1987), Nur Al-Mantiq Wal-Ma’rifah Villeurbanne [25] Steinberg, R., Lectures on Chevalley groups, (1967), Yale University [26] Zil’ber, B.I., Groups and rings whose theory is categorical, Fund. math., 55, 173-188, (1977) · Zbl 0363.02060 [27] Zil’ber, B.I., Some model theory of simple algebraic groups over algebraically closed fields, Colloq. math., 48, 173-180, (1984) · Zbl 0567.20030
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