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 ...
, 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 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 ...
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
finite-dimensional objects. The striking similarities between groups of finite Morley rank and finite groups are an object of active research.
*All
finite groups 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.
...
over
algebraically closed fields 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 coor ...
as
algebraic sets.
* 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 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 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.
...
s over
algebraically closed fields. 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
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 cl ...
. (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
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*
* (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