200:'s trichotomy conjecture. Cherlin posed the question for all ω-stable simple groups, but remarked that even the case of groups of finite Morley rank seemed hard.
267:) showed that an infinite group of finite Morley rank is either an algebraic group over an algebraically closed field of characteristic 2, or has finite 2-rank.
467:
17:
358:, NATO ASI Series C: Mathematical and Physical Sciences, vol. 517, Dordrecht: Kluwer Academic Publishers, pp. 341–366
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objects. The striking similarities between groups of finite Morley rank and finite groups are an object of active research.
514:, Mathematical Surveys and Monographs, vol. 87, Providence, RI: American Mathematical Society, pp. xiv+129,
605:
527:
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109:; therefore groups of finite Morley rank are stable groups. Groups of finite Morley rank behave in certain ways like
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465:(2010), "Review of "Simple groups of finite Morley rank" by T. Altinel, A. V. Borovik and G. Cherlin",
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Cherlin proved that a simple group of Morley rank 3 is either a bad group or isomorphic to PSL
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connected groups of finite Morley rank all of whose proper connected definable subgroups are
35:
619:"Группы и кольца, теория которых категорична (Groups and rings whose theory is categorical)"
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354:(1998), "Tame groups of odd and even type", in Carter, R. W.; Saxl, J. (eds.),
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Tuna Altinel, Alexandre V. Borovik, and Gregory
Cherlin (
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A number of special cases of this conjecture have been proved; for example:
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375:, Oxford Logic Guides, vol. 26, New York: Oxford University Press,
315:, Mathematical Surveys and Monographs, vol. 145, Providence, R.I.:
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281:; Borovik, Alexandre; Cherlin, Gregory (1997), "Groups of mixed type",
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if it has no definable subgroups of finite index other than itself.)
311:
Altinel, Tuna; Borovik, Alexandre V.; Cherlin, Gregory (2008),
207:’s program of transferring methods used in classification of
155:, are stable. Free groups on more than one generator are not
241:
Cherlin proved that a connected rank 2 group is solvable.
545:
Scanlon, Thomas (2002), "Review of "Stable groups"",
310:
264:
440:Cherlin, G. (1979), "Groups of small Morley rank",
399:"The Bender method in groups of finite Morley rank"
368:
61:. An important class of examples is provided by
652:
572:Diophantine Geometry over Groups VIII: Stability
163:
203:Progress towards this conjecture has followed
30:For stable groups in finite group theory, see
468:Bulletin of the American Mathematical Society
211:. One possible source of counterexamples is
34:. For stable groups in homotopy theory, see
613:
541:(Translated from the 1987 French original.)
196:. The conjecture would have followed from
181:
356:Algebraic Groups and their Representations
101:. It follows from the definition that the
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234:Any connected group of Morley rank 1 is
130:have finite Morley rank, equal to their
120:have finite Morley rank, in fact rank 0.
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252:) for some algebraically closed field
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184:, suggests that infinite (ω-stable)
141:
105:of a group of finite Morley rank is
313:Simple groups of finite Morley rank
24:
25:
682:
57:that is stable in the sense of
600:, Cambridge University Press,
418:10.1016/j.jalgebra.2005.10.009
13:
1:
560:10.1090/S0273-0979-02-00953-9
494:"Group of finite Morley rank"
482:10.1090/S0273-0979-10-01287-5
317:American Mathematical Society
271:
164:The Cherlin–Zilber conjecture
455:10.1016/0003-4843(79)90019-6
371:Groups of Finite Morley Rank
63:groups of finite Morley rank
7:
596:Wagner, Frank Olaf (1997),
499:Encyclopedia of Mathematics
194:algebraically closed fields
128:algebraically closed fields
76:group of finite Morley rank
68:
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687:
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397:Burdges, Jeffrey (2007),
170:Cherlin–Zilber conjecture
18:Cherlin–Zilber conjecture
636:10.4064/fm-95-3-173-188
174:algebraicity conjecture
27:Concept in model theory
547:Bull. Amer. Math. Soc.
510:Poizat, Bruno (2001),
296:10.1006/jabr.1996.6950
85:such that the formula
40:direct limit of groups
661:Infinite group theory
223:. (A group is called
148:, and more generally
36:stable homotopy group
671:Properties of groups
209:finite simple groups
590:2006math......9096S
463:Macpherson, Dugald
176:), due to Gregory
111:finite-dimensional
336:978-0-8218-4305-5
172:(also called the
153:hyperbolic groups
16:(Redirected from
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144:showed that
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51:stable group
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47:model theory
44:
629:: 173–188,
260:interprets.
188:are simple
157:superstable
146:free groups
142:Sela (2006)
95:Morley rank
93:has finite
655:Categories
568:Sela, Zlil
406:J. Algebra
365:Nesin, Ali
283:J. Algebra
272:References
217:nonsoluble
213:bad groups
180:and Boris
504:EMS Press
492:(2001) ,
225:connected
221:nilpotent
132:dimension
617:(1977),
570:(2006),
367:(1994),
107:ω-stable
69:Examples
645:0441720
586:Bibcode
538:1827833
434:9031997
426:2320445
391:1321141
345:2400564
305:1452677
236:abelian
205:Borovik
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198:Zilber
103:theory
576:arXiv
430:S2CID
402:(PDF)
256:that
192:over
126:over
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602:ISBN
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