View source for Stable group

{{Short description|Concept in model theory}}
{{about||stable groups in finite group theory|p-stable group|stable groups in homotopy theory|stable homotopy group|and|direct limit of groups}}
In [[model theory]], a '''stable group''' is a [[group (mathematics)|group]] that is stable in the sense of [[stable theory|stability theory]]. 
An important class of examples is provided by '''groups of finite Morley rank''' (see below).

==Examples==
*A '''group of finite Morley rank''' is an abstract [[group (mathematics)|group]] ''G'' such that the formula ''x''&nbsp;=&nbsp;''x'' has finite [[Morley rank]] for the model ''G''. It follows from the definition that the [[theory (logic)|theory]] of a group of finite Morley rank is [[ω-stable]]; therefore groups of finite Morley rank are stable groups. Groups of finite Morley rank behave in certain ways like [[dimension (mathematics)|finite-dimensional]] objects. The striking similarities between groups of finite Morley rank and finite groups are an object of active research.
*All [[finite group]]s have finite Morley rank, in fact rank 0.
*[[Algebraic groups]] over [[algebraically closed field]]s have finite Morley rank, equal to their [[Dimension of an algebraic variety|dimension]] as [[algebraic set]]s.
*{{harvtxt|Sela|2006}} showed that [[free group]]s, and more generally [[torsion-free group|torsion-free]] [[hyperbolic group]]s, are stable. Free groups on more than one generator are not [[superstable]].

==The Cherlin–Zilber conjecture==
The '''Cherlin–Zilber conjecture''' (also called the '''algebraicity conjecture'''), due to Gregory {{harvtxt|Cherlin|1979}} and Boris {{harvtxt|Zil'ber|1977}}, suggests that infinite (ω-stable) [[simple groups]] are simple [[algebraic group]]s over [[algebraically closed field]]s.  The conjecture would have followed from [[Boris Zilber|Zilber]]'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.

Progress towards this conjecture has followed [[Alexandre Borovik|Borovik]]’s program of transferring methods used in classification of [[finite simple groups]]. One possible source of counterexamples is '''bad groups''': [[soluble group|nonsoluble]] connected groups of finite Morley rank all of whose proper connected definable subgroups are [[Nilpotent group|nilpotent]]. (A group is called '''connected''' if it has no definable subgroups of finite index other than itself.)

A number of special cases of this conjecture have been proved; for example:
*Any connected group of Morley rank 1 is [[abelian group|abelian]].
*Cherlin proved that a connected rank 2 group is solvable.
*Cherlin proved that a simple group of Morley rank 3 is either a bad group or isomorphic to PSL<sub>2</sub>(''K'') for some algebraically closed field ''K'' that ''G'' interprets.
*{{harvs|txt | last1=Altinel | first1=Tuna | last2=Borovik | first2=Alexandre V. | last3=Cherlin | first3=Gregory | title=Simple groups of finite Morley rank | publisher=[[American Mathematical Society]] | location=Providence, R.I. | series=Mathematical Surveys and Monographs | isbn=978-0-8218-4305-5 | mr=2400564 | year=2008 | volume=145}} 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.

==References==
*{{citation
|mr=1452677|last1=Altinel|first1=Tuna|author1link = Tuna Altinel|last2=Borovik|first2=Alexandre|last3=Cherlin|first3= Gregory |title=Groups of mixed type|journal=J. Algebra|volume=192|year=1997|issue=2|pages=524–571|doi=10.1006/jabr.1996.6950 |ref=none|doi-access=free}}
*{{Citation | last1=Altinel | first1=Tuna | last2=Borovik | first2=Alexandre V. | last3=Cherlin | first3=Gregory | title=Simple groups of finite Morley rank | publisher=[[American Mathematical Society]] | location=Providence, R.I. | series=Mathematical Surveys and Monographs | isbn=978-0-8218-4305-5 | mr=2400564 | year=2008 | volume=145 | doi=10.1090/surv/145| doi-access=free }}
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* {{citation|first1= A. V.|last1= Borovik|first2= Ali|last2= Nesin|authorlink2 = Ali Nesin|title= Groups of Finite Morley Rank|publisher= Oxford University Press|year= 1994|mr= 1321141|series= Oxford Logic Guides|volume= 26|place= New York|isbn= 0-19-853445-0|ref= none|url-access= registration|url= https://archive.org/details/groupsoffinitemo0000boro}}
*{{citation|url=http://www.math.rutgers.edu/~burdges/BenderMethod.pdf |first=Jeffrey |last=Burdges|title= The Bender method in groups of finite Morley rank|mr=2320445 |journal=J. Algebra|volume=312|year=2007|issue= 1|pages=33–55|doi=10.1016/j.jalgebra.2005.10.009|s2cid=9031997 |ref=none}}
*{{citation|first=G.|last= Cherlin|title=Groups of small Morley rank|journal=Ann. Math. Logic |volume= 17  |year=1979|issue=1–2|pages= 1–28|doi=10.1016/0003-4843(79)90019-6|doi-access=free}}
*{{Citation | last1=Macpherson | first1=Dugald | author1-link=Dugald Macpherson | title=Review of "Simple groups of finite Morley rank" by T. Altinel, A. V. Borovik and G. Cherlin | doi=10.1090/S0273-0979-10-01287-5  | year=2010 | journal=[[Bulletin of the American Mathematical Society]] | volume=47 | issue=4 | pages=729–734|ref=none| doi-access=free }}
*{{springer|id=g/g110270|title=Group of finite Morley rank|first=Anand |last=Pillay|authorlink = Anand Pillay|ref=none}}
*{{citation|mr=1827833 |last=Poizat|first= Bruno |title=Stable groups |series= Mathematical Surveys and Monographs|volume= 87 |publisher=American Mathematical Society|publication-place= Providence, RI|year= 2001|pages= xiv+129| isbn= 0-8218-2685-9 |doi=10.1090/surv/087|ref=none}} (Translated from the 1987 French original.)
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*{{Citation | last1=Sela | first1=Zlil | author-link = Zlil Sela|title=Diophantine Geometry over Groups VIII: Stability | arxiv=math/0609096 | year=2006| bibcode=2006math......9096S }}
*{{citation|first=Frank Olaf |last=Wagner |year=1997 |publisher=Cambridge University Press |isbn= 0-521-59839-7|title=Stable groups|ref=none}}
*{{citation|first=B. I.|last= Zil'ber|author-link=Boris Zilber|title=Группы и кольца, теория которых категорична (Groups and rings whose theory is categorical)|journal=Fundam. Math.|volume= 95|year=1977|pages=173–188 |doi= 10.4064/fm-95-3-173-188|url=http://matwbn.icm.edu.pl/tresc.php?wyd=1&tom=95&jez=|mr=0441720 |doi-access=free}}

[[Category:Infinite group theory]]
[[Category:Model theory]]
[[Category:Properties of groups]]