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Semisimple stable and superstable groups
1989
Annals of Pure and Applied Logic
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A subgroup H of a group G is type-dejiruzbk or (/\-definable) if universe of H is the intersection of a family of at most IL1 definable subsets G) and the CC unction of the & define a subgroup in every ...
In fact, most of our type-definable arise in this way. 0 hand, Lascar and Berline in [6] at, again when G is stable, the intersection of any family of deF$able al to the intersection of at most IL1 definable ...
Cl
Let H be a type-definable subgroup of a 1 TI+-saturated stable group G. ...
doi:10.1016/0168-0072(89)90056-0
fatcat:uhnidgfahfd5zoizotc2dfu2wi
Stable groups and expansions of (Z,+,0)
[article]
2017
arXiv
pre-print
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In particular, this answers a question of Marker on stable expansions of the group of integers by sets definable in Presburger arithmetic. ...
We show that if G is a sufficiently saturated stable group of finite weight with no infinite, infinite-index, chains of definable subgroups, then G is superstable of finite U-rank. ...
We are grateful to Rizos Sklinos and Erik Walsberg for their comments on an earlier draft. ...
arXiv:1601.05692v3
fatcat:3654u6px2zeadpagjjmwamvzkm
ON MAXIMAL STABLE QUOTIENTS OF DEFINABLE GROUPS IN NIP THEORIES
2018
Journal of Symbolic Logic (JSL)
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AbstractFor G a group definable in a saturated model of a NIP theory T, we prove that there is a smallest type-definable subgroup H of G such that the quotient G / H is stable. ...
This generalizes the existence of G00, the smallest type-definable subgroup of G of bounded index. ...
Suppose T has N IP , and let G be a definable group (over ∅). Let H be a type-definable subgroup of G such that G/H is stable. ...
doi:10.1017/jsl.2017.26
fatcat:mnngwjv4dngexbprprdnknrhma
A note on superstable groups
2005
Journal of Symbolic Logic (JSL)
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It is proved that all groups of finite U-rank that have the descending chain condition on definable subgroups are totally transcendental. ...
A corollary is that any stable group that is definable in an o -minimal structure is totally transcendental of finite Morley rank. ...
Poizat's theorem [8] that every type-definable subgroup of a stable group is the intersection of definable subgroups, together with the d.c.c., implies that H is a definable subgroup of G/H n . ...
doi:10.2178/jsl/1120224734
fatcat:fqikxdbalzhkxerdtsvhyygm5m
On maximal stable quotients of definable groups in NIP theories
[article]
2016
arXiv
pre-print
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For G a group definable in an NIP theory we prove that there is a smallest type-definable subgroup H of G such that the quotient G/H is stable. ...
This generalizes the existence of G^00, the smallest type-definable subgroup of G of bounded index. ...
Suppose T has NIP , and let G be a definable group (over ∅). Let H be a type-definable subgroup of G such that G/H is stable. ...
arXiv:1606.01182v1
fatcat:k7y7l5rtibajtcnefijchroqzy
On Variants of CM-triviality
[article]
2012
arXiv
pre-print
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We show that, under this condition, a stable field is internal to the family, and a group of finite Lascar rank has a normal nilpotent subgroup such that the quotient is almost internal to the family. ...
We introduce a generalization of CM-triviality relative to a fixed invariant collection of partial types, in analogy to the Canonical Base Property defined by Pillay, Ziegler and Chatzidakis which generalizes ...
Hrushovski and Pillay have shown [9] that in a stable one-based group, definable sets are finite unions of cosets of acl eq (∅)-definable subgroups. Furthermore, the group is abelian-by-finite. ...
arXiv:1205.6916v2
fatcat:km5zqtzhwjabrfpzwqmvq3zg3m
On variants of CM-triviality
2012
Fundamenta Mathematicae
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We show that, under this condition, a stable field is internal to the family, and a group of finite Lascar rank has a normal nilpotent subgroup such that the quotient is almost internal to the family. ...
We introduce a generalization of CM-triviality relative to a fixed invariant collection of partial types, in analogy to the Canonical Base Property defined by Pillay, Ziegler and Chatzidakis which generalizes ...
Hrushovski and Pillay have shown [9] that in a stable one-based group, definable sets are finite unions of cosets of acl eq (∅)-definable subgroups. Furthermore, the group is abelian-by-finite. ...
doi:10.4064/fm219-3-4
fatcat:pybsajulyzhldewjxyzs5cnyqm
Book Review: Stable groups
2002
Bulletin of the American Mathematical Society
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This result is followed by a series of propositions on descending chain conditions on subgroups of stable groups. ...
We define G 0 (φ) to be the intersection of all subgroups of G of the form H a := {g ∈ G : φ(g, a)} of finite index in G. For a stable group G, the group G 0 (φ) is necessarily definable. ...
Thomas Scanlon University of California, Berkeley E-mail address: scanlon@math.berkeley.edu ...
doi:10.1090/s0273-0979-02-00953-9
fatcat:bka54xadg5eavlvx47pbipt2ea
DEFINABLE GROUPS IN DCFA
2019
Revista de Matemática: Teoría y Aplicaciones
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Then, we study one-basedeness, stability and stable embeddability of abelian definable groups. ...
In this paper we study definable abelian groups in a model of DCFA. First we prove that such a group is embeddable on an algebraic group. ...
By 1-basedness and quantifier-free ω-stability, we know that if X ⊂ A ♯ is quantifier-free definable, then X is a Boolean combination of cosets of quantifier-free definable subgroups of A ♯ . ...
doi:10.15517/rmta.v26i2.33639
fatcat:uzkkiqb4nvfm5e2noryooyy5p4
Complete reducibility: Variations on a theme of Serre
[article]
2021
arXiv
pre-print
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By results of Serre and Bate–Martin–Röhrle, the usual notion of G-complete reducibility can be re-framed as a property of an action of a group on the spherical building of the identity component of G. ...
In this note, we unify and extend various concepts in the area of G-complete reducibility, where G is a reductive algebraic group. ...
Acknowledgements: We would like to thank Michael Bate for helpful discussions on the material of this note. ...
arXiv:2004.14604v3
fatcat:tkpmsvy535g2tisbufbfsraji4
Complete reducibility: variations on a theme of Serre
2021
Manuscripta mathematica
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By results of Serre and Bate–Martin–Röhrle, the usual notion of G-complete reducibility can be re-framed as a property of an action of a group on the spherical building of the identity component of G. ...
AbstractIn this note, we unify and extend various concepts in the area of G-complete reducibility, where G is a reductive algebraic group. ...
To view a copy of this licence, visit http://creativecommons.org/ licenses/by/4.0/. ...
doi:10.1007/s00229-021-01318-2
fatcat:cpuffrdarnfo3h6qxpvnzci3zy
Book Review: Simple groups of finite Morley rank
2010
Bulletin of the American Mathematical Society
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In the mid-1990s a view was consolidated that this project falls into four cases of different flavour: even type, mixed type, odd type, and degenerate type. ...
Morley showed that in an uncountably categorical theory, all definable sets have ordinal-valued Morley rank. In particular, such theories are stable (in fact, ω-stable). ...
points of an algebraic group defined over a stable field K. ...
doi:10.1090/s0273-0979-10-01287-5
fatcat:7wkxy6qbw5fc5pp2vrjmhd7xue
Page 4133 of Mathematical Reviews Vol. , Issue 92h
[page]
1992
Mathematical Reviews
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A series of applications of type-definable subgroups of a stable group is given. ...
(PL-PAW)
On type definable subgroups of a stable group.
Notre Dame J. Formal Logic 32 (1991), no. 2, 173-187. ...
Definable Groups in DCFA
[article]
2019
arXiv
pre-print
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First we prove that such a group is embeds on an algebraic group. Then we study 1-basedeness, stability and stable embeddability of abelian definable groups. ...
Hrushovski proved that the theory of difference-differential fields of characteristic zero has a model-companion. We denote it DCFA. In this paper we study definable groups in a model of DCFA. ...
with a ′ over M , b is interalgebraic with b ′ over M and c is interalgebraic with c ′ over M Then there is a type-definable over M subgroup G 1 of bounded index in G, and a type-definable over M subgroup ...
arXiv:1904.11586v1
fatcat:zwuzq2mwvfftnm4cwvohv4ueem
Model Theory and Groups
[article]
2021
arXiv
pre-print
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This is a largely expository paper about how groups arise or are of interest in model theory. ...
Included are the following topics: classifying groups definable in specific structures or theories and the relation to algebraic groups, groups definable in stable, simple and NIP theories, definable compactifications ...
Stable groups Stable group theory is an adaptation of the above machinery to the context of definable homogeneous spaces (a transitive definable action of a definable group G on a definable set X), although ...
arXiv:2109.03911v1
fatcat:lkzf7qgd65e75kb2ng2zmvanmq
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