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
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 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
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 ...
''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
A theory is a rational type of abstract thinking about a phenomenon, or the results of such thinking. The process of contemplative and rational thinking is often associated with such processes as observational study or research. Theories may be s ...
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
In mathematics, an algebraic group is an algebraic variety endowed with a group structure which is compatible with its structure as an algebraic variety. Thus the study of algebraic groups belongs both to algebraic geometry and group theory.
M ...
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
In physics and mathematics, the dimension of a Space (mathematics), mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any Point (geometry), point within it. Thus, a Line (geometry), lin ...
as
algebraic set
Algebraic may refer to any subject related to algebra in mathematics and related branches like algebraic number theory and algebraic topology. The word algebra itself has several meanings.
Algebraic may also refer to:
* Algebraic data type, a dat ...
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''−1' ...
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 abstra ...
s, are stable. Free groups on more than one generator are not
superstable
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 ...
.
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 which 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
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'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
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 ...
. 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 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 was introduced by Benjamin Peirce in the context of his work on the class ...
. (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.
*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