In
model theory
In mathematical logic, model theory is the study of the relationship between formal theories (a collection of sentences in a formal language expressing statements about a mathematical structure), and their models (those structures in which the s ...
, a stable group is a
group 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 diffe ...
.
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 ''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 of a theory, generalizing the notion of dimension in algebraic geometry.
Definition
Fix a theory ''T'' with a model ''M''. The Morley rank ...
for the model ''G''. It follows from the definition that the
theory of a group of finite Morley rank is
ω-stable
In the mathematical field of model theory, a complete theory is called stable if it does not have too many types. One goal of classification theory is to divide all complete theories into those whose models can be classified and those whose model ...
; 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 disti ...
objects. The striking similarities between groups of finite Morley rank and finite groups are an object of active research.
*All
finite group
Finite is the opposite of infinite. It may refer to:
* Finite number (disambiguation)
* Finite set, a set whose cardinality (number of elements) is some natural number
* Finite verb, a verb form that has a subject, usually being inflected or marked ...
s have finite Morley rank, in fact rank 0.
*
Algebraic groups 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 .
Examples
As an example, the field of real numbers is not algebraically closed, because ...
s have finite Morley rank, equal to their
dimension as
algebraic sets.
* showed that
free groups, and more generally
torsion-free hyperbolic groups, 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 and Boris , suggests that infinite (ω-stable)
simple groups
SIMPLE Group Limited is a conglomeration of separately run companies that each has its core area in International Consulting. The core business areas are Legal Services, Fiduciary Activities, Banking Intermediation and Corporate Service.
The da ...
are simple
algebraic groups 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 .
Examples
As an example, the field of real numbers is not algebraically closed, because ...
s. The conjecture would have followed from
Zilber Zilber ( yi, זילבּער, russian: Зильбер) is a surname and a variation of ''Silber''. Notable people with the surname include:
* Ariel Zilber ( אריאל זילבר; born 1943), Israeli musical artist
* Belu Zilber (1901–1978), Rom ...
'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. One possible source of counterexamples is bad groups:
nonsoluble connected groups of finite Morley rank all of whose proper connected definable subgroups are
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
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 grou ...
.
*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
2(''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 , url=http://matwbn.icm.edu.pl/tresc.php?wyd=1&tom=95&jez=, mr=0441720
Infinite group theory
Model theory
Properties of groups