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Semisimple stable and superstable groups

J.T. Baldwin, A. Pillay
1989 Annals of Pure and Applied Logic  
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]

Gabriel Conant, Anand Pillay
2017 arXiv   pre-print
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

MIKE HASKEL, ANAND PILLAY
2018 Journal of Symbolic Logic (JSL)  
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

Jerry Gagelman
2005 Journal of Symbolic Logic (JSL)  
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]

Mike Haskel, Anand Pillay
2016 arXiv   pre-print
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]

Thomas Blossier, Frank Olaf Wagner
2012 arXiv   pre-print
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

Thomas Blossier, Amador Martin-Pizarro, Frank O. Wagner
2012 Fundamenta Mathematicae  
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

Thomas Scanlon
2002 Bulletin of the American Mathematical Society  
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

RONALD BUSTAMANTE MEDINA
2019 Revista de Matemática: Teoría y Aplicaciones  
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]

Maike Gruchot, Alastair Litterick, Gerhard Roehrle
2021 arXiv   pre-print
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

Maike Gruchot, Alastair Litterick, Gerhard Röhrle
2021 Manuscripta mathematica  
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

Dugald Macpherson
2010 Bulletin of the American Mathematical Society  
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  
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]

Ronald F. Bustamante Medina
2019 arXiv   pre-print
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]

Anand Pillay
2021 arXiv   pre-print
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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