Stable group
Encyclopedia
In model theory
, a stable group is a group that is stable in the sense of stability theory
.
An important class of examples is provided by groups of finite Morley rank (see below).
s over algebraically closed field
s. The conjecture would have followed from Zilber 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:
Model theory
In mathematics, model theory is the study of mathematical structures using tools from mathematical logic....
, a stable group is a group that is stable in the sense of stability theory
Stable theory
In 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 models are too complicated to classify, and to classify all models in the cases...
.
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 groupGroup (mathematics)In mathematics, a group is an algebraic structure consisting of a set together with an operation that combines any two of its elements to form a third element. To qualify as a group, the set and the operation must satisfy a few conditions called group axioms, namely closure, associativity, identity...
G such that the formula x = x has finite Morley rankMorley rankIn 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...
for the model G. It follows from the definition that the 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 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 over algebraically closed fields 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 are simple algebraic groupAlgebraic group
In algebraic geometry, an algebraic group is a group that is an algebraic variety, such that the multiplication and inverse are given by regular functions on the variety...
s over algebraically closed field
Algebraically closed field
In mathematics, a field F is said to be algebraically closed if every polynomial with one variable of degree at least 1, with coefficients in F, has a root in F.-Examples:...
s. The conjecture would have followed from Zilber 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
Alexandre Borovik
Alexandre Borovik is Professor of Pure Mathematics at the University of Manchester, United Kingdom, where he has been working for the past 20 years. His principal research lies in algebra, model theory, and combinatorics---topics on which he published several monographs and a number of papers...
’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
Nilpotent group
In mathematics, more specifically in the field of group theory, a nilpotent group is a group that is "almost abelian". This idea is motivated by the fact that nilpotent groups are solvable, and for finite nilpotent groups, two elements having relatively prime orders must commute...
, (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 PSL2(K) for some algebraicially 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.