model theory In mathematical logic, model theory is the study of the relationship between theory (mathematical logic), formal theories (a collection of Sentence (mathematical logic), sentences in a formal language expressing statements about a Structure (mat ...

, a stable group is a

group A group is a number of persons or things that are located, gathered, or classed together. Groups of people * Cultural group, a group whose members share the same cultural identity * Ethnic group, a group whose members share the same ethnic iden ...

that is stable in the sense of

stability theory In mathematics, stability theory addresses the stability of solutions of differential equations and of trajectories of dynamical systems under small perturbations of initial conditions. The heat equation, for example, is a stable partial differ ...

. 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

''G'' such that the formula ''x'' = ''x'' has finite

Morley rank In mathematical logic, Morley rank, introduced by , is a means of measuring the size of a subset of a model theory, model of a theory (logic), theory, generalizing the notion of dimension in algebraic geometry. Definition Fix a theory ''T'' with a ...

for the model ''G''. It follows from the definition that the

theory A theory is a systematic and rational form of abstract thinking about a phenomenon, or the conclusions derived from such thinking. It involves contemplative and logical reasoning, often supported by processes such as observation, experimentation, ...

of a group of finite Morley rank is

ω-stable In the mathematical field of model theory, a theory is called stable if it satisfies certain combinatorial restrictions on its complexity. Stable theories are rooted in the proof of Morley's categoricity theorem and were extensively studied as p ...

; therefore groups of finite Morley rank are stable groups. Groups of finite Morley rank behave in certain ways like

finite-dimensional In mathematics, the dimension of a vector space ''V'' is the cardinality (i.e., the number of vectors) of a basis of ''V'' over its base field. p. 44, §2.36 It is sometimes called Hamel dimension (after Georg Hamel) or algebraic dimension to d ...

objects. The striking similarities between groups of finite Morley rank and finite groups are an object of active research. *All

finite group In abstract algebra, a finite group is a group whose underlying set is finite. Finite groups often arise when considering symmetry of mathematical or physical objects, when those objects admit just a finite number of structure-preserving tra ...

s have finite Morley rank, in fact rank 0. *

Algebraic groups In mathematics, an algebraic group is an algebraic variety endowed with a group structure that is compatible with its structure as an algebraic variety. Thus the study of algebraic groups belongs both to algebraic geometry and group theory. Many ...

over

algebraically closed field In mathematics, a field is algebraically closed if every non-constant polynomial in (the univariate polynomial ring with coefficients in ) has a root in . In other words, a field is algebraically closed if the fundamental theorem of algebra ...

s have finite Morley rank, equal to their

dimension In physics and mathematics, the dimension of a mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any point within it. Thus, a line has a dimension of one (1D) because only one coo ...

algebraic set Algebraic varieties are the central objects of study in algebraic geometry, a sub-field of mathematics. Classically, an algebraic variety is defined as the set of solutions of a system of polynomial equations over the real or complex numbers. ...

s. * showed that

free group In mathematics, the free group ''F'S'' over a given set ''S'' consists of all words that can be built from members of ''S'', considering two words to be different unless their equality follows from the group axioms (e.g. ''st'' = ''suu''− ...

s, and more generally torsion-free

hyperbolic group In group theory, more precisely in geometric group theory, a hyperbolic group, also known as a ''word hyperbolic group'' or ''Gromov hyperbolic group'', is a finitely generated group equipped with a word metric satisfying certain properties abs ...

s, are stable. Free groups on more than one generator are not

superstable In the mathematical field of model theory, a theory is called stable if it satisfies certain combinatorial restrictions on its complexity. Stable theories are rooted in the proof of Morley's categoricity theorem and were extensively studied as p ...

The Cherlin–Zilber conjecture

The Cherlin–Zilber conjecture (also called the algebraicity conjecture), due to Gregory and Boris , suggests that infinite (ω-stable) simple groups are simple

algebraic group In mathematics, an algebraic group is an algebraic variety endowed with a group structure that is compatible with its structure as an algebraic variety. Thus the study of algebraic groups belongs both to algebraic geometry and group theory. Man ...

s over

s. The conjecture would have followed from

Zilber Zilber (, ) is a surname and a variation of ''Silber (disambiguation), Silber''. Notable people with the surname include: * Ariel Zilber (:he:אריאל זילבר, אריאל זילבר; born 1943), Israeli musical artist * Belu Zilber (1901–197 ...

'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 Borovik’s program of transferring methods used in classification of

finite simple groups In mathematics, the classification of finite simple groups states that every finite simple group is cyclic, or alternating, or in one of 16 families of groups of Lie type, or one of 26 sporadic groups. The list below gives all finite simple gr ...

. One possible source of counterexamples is bad groups: nonsoluble connected groups of finite Morley rank all of whose proper connected definable subgroups are

nilpotent In mathematics, an element x of a ring (mathematics), ring R is called nilpotent if there exists some positive integer n, called the index (or sometimes the degree), such that x^n=0. The term, along with its sister Idempotent (ring theory), idem ...

. (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 Abelian may refer to: Mathematics Group theory * Abelian group, a group in which the binary operation is commutative ** Category of abelian groups (Ab), has abelian groups as objects and group homomorphisms as morphisms * Metabelian group, a group ...

. *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 PSL₂(''K'') for some algebraically closed field ''K'' that ''G'' interprets. * 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

* * * * * * * * * (Translated from the 1987 French original.) * * * *{{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 Infinite group theory Model theory Properties of groups