Results 1  10
of
14
Superrosy dependent groups having finitely satisfiable generics
"... We develop a basic theory of rosy groups and we study groups of small Uþrank satisfying NIP and having finitely satisfiable generics: Uþrank 1 implies that the group is abelianbyfinite, Uþrank 2 implies that the group is solvablebyfinite, Uþrank 2, and not being nilpotentbyfinite implies t ..."
Abstract

Cited by 5 (2 self)
 Add to MetaCart
We develop a basic theory of rosy groups and we study groups of small Uþrank satisfying NIP and having finitely satisfiable generics: Uþrank 1 implies that the group is abelianbyfinite, Uþrank 2 implies that the group is solvablebyfinite, Uþrank 2, and not being nilpotentbyfinite implies the existence of an interpretable algebraically closed field.
Groups In Which Commutativity Is A Transitive Relation
 Department of Mathematics, Statistics, and Computer Science, University of WisconsinStout, Menomonie, WI
, 1998
"... . We investigate the structure of groups in which commutativity is a transitive relation on nonidentity elements (CT groups). A detailed study of locally finite, polycyclic, and torsionfree solvable CT groups is carried out. Other topics include fixedpointfree groups of automorphisms of abelia ..."
Abstract

Cited by 2 (1 self)
 Add to MetaCart
. We investigate the structure of groups in which commutativity is a transitive relation on nonidentity elements (CT groups). A detailed study of locally finite, polycyclic, and torsionfree solvable CT groups is carried out. Other topics include fixedpointfree groups of automorphisms of abelian torsion groups and their cohomology groups. 1. Introduction Definition. A group G is called a CT group if [x; y] = 1 and [y; z] = 1 imply that [x; z] = 1, for all nontrivial elements x; y; z 2 G. In other words, the relation of commutativity is transitive on the set Gnf1g. In other papers, CT groups have been called CAgroups since the centralizer of every nonidentity element is abelian. Obvious examples of CT groups include abelian groups and free groups; the Tarski groups, which are simple groups with all proper subgroups cyclic, are also CT groups (see [9], Theorem 28.3). This shows how complicated the structure of CT groups can be. Finite CT groups were first studied by Weisne...
IDEALS IN GROUP ALGEBRAS OF SIMPLE LOCALLY FINITE GROUPS OF 1TYPE
"... Let K be any field of characteristic zero. We show that there are at least four ideals in the group algebra KG of every simple locally finite group G of 1type, thus providing the final step in solving an old question of I. Kaplansky’s for locally finite groups. We also determine the ideal lattice i ..."
Abstract

Cited by 1 (0 self)
 Add to MetaCart
Let K be any field of characteristic zero. We show that there are at least four ideals in the group algebra KG of every simple locally finite group G of 1type, thus providing the final step in solving an old question of I. Kaplansky’s for locally finite groups. We also determine the ideal lattice in KG for those 1type groups G which are a direct limit of finite direct products of alternating groups. 1. Introduction. It is an old question due to I. Kaplansky [K] for which groups G and which fields K the augmentation ideal ω(KG) is the only nonzero proper twosided ideal in KG. Every such group G must necessarily be simple. Kaplansky’s question was the starting point for a research programme begun by A.E.
Some model theory of Polish structures
"... We introduce a notion of Polish structure and, in doing so, provide a setting which allows the application of ideas and techniques from model theory, descriptive set theory, topology and the theory of profnite groups. We define a topological notion of independence in Polish structures and prove that ..."
Abstract

Cited by 1 (1 self)
 Add to MetaCart
We introduce a notion of Polish structure and, in doing so, provide a setting which allows the application of ideas and techniques from model theory, descriptive set theory, topology and the theory of profnite groups. We define a topological notion of independence in Polish structures and prove that it has some nice properties. Using this notion, we prove counterparts of some basic results from geometric stability theory in the context of small Polish structures. Then we prove some structural theorems about compact groups regarded as Polish structures: each small, nmstable compact Ggroup is solvablebyfinite; each small compact Ggroup of finite NMrank is nilpotentbyfinite. Examples of small Polish structures and groups are also given.
PERIODIC GROUPS COVERED BY TRANSITIVE SUBGROUPS OF FINITARY PERMUTATIONS OR BY IRREDUCIBLE SUBGROUPS OF FINITARY TRANSFORMATIONS
"... Abstract. Let X be either the class of all transitive groups of finitary permutations, or the class of all periodic irreducible finitary linear groups. We show that almost primitive Xgroups are countably recognizable, while totally imprimitive Xgroups are in general not countably recognizable. In ..."
Abstract
 Add to MetaCart
Abstract. Let X be either the class of all transitive groups of finitary permutations, or the class of all periodic irreducible finitary linear groups. We show that almost primitive Xgroups are countably recognizable, while totally imprimitive Xgroups are in general not countably recognizable. In addition we derive a structure theorem for groups all of whose countable subsets are contained in totally imprimitive Xsubgroups. It turns out that totally imprimitive pgroups in the class X are countably recognizable. 1.
FROBENIUS COMPLEMENTS OF EXPONENT DIVIDING 2 m · 9
, 2007
"... Abstract. We show that every group of exponent 2 m · 3 n (m, n ∈ N, n ≤ 2) that acts freely on some abelian group is finite. 1. Results Let V be a group, and let G be a group of automorphisms of V. We say that G acts freely on V if v g ̸ = v for all v ∈ V \ {1} and g ∈ G \ {1}. In the literature thi ..."
Abstract
 Add to MetaCart
Abstract. We show that every group of exponent 2 m · 3 n (m, n ∈ N, n ≤ 2) that acts freely on some abelian group is finite. 1. Results Let V be a group, and let G be a group of automorphisms of V. We say that G acts freely on V if v g ̸ = v for all v ∈ V \ {1} and g ∈ G \ {1}. In the literature this concept is also often called regular or fixedpointfree action of G on V. We consider free actions of groups of finite exponent. In [1] the first author proved that groups of exponent 5 that act freely on abelian groups are finite. In the present note we show the following. Theorem 1.1. Let V be an abelian group, and let G be a group of automorphisms of V. If G has exponent 2 m · 3 n for 0 ≤ m and 0 ≤ n ≤ 2 and G acts freely on V, then G is finite. Every finite group that acts freely on an abelian group is isomorphic to a Frobenius complement in some finite Frobenius group (see Lemma 2.6). Let G be as in Theorem 1.1. By the classification of finite Frobenius complements (see [6]) the factor of G by its maximal normal 3subgroup is isomorphic to a cyclic 2group, a generalized quaternion group, SL(2, 3), or the binary octahedral group of size 48. Corollary 1.2. Let F be a nearfield whose multiplicative group has exponent 2 m · 3 n for 0 ≤ m and 0 ≤ n ≤ 2. Then either F  ∈ {2 2, 3 2, 5 2, 7 2, 17 2} or F is a finite field of prime order. We note that there exist nearfields of orders 3 2, 5 2, 7 2, 17 2 that are not fields. Every zerosymmetric nearring with 1, whose elements satisfy xk = x for a fixed integer k> 1, is a subdirect product of nearfields satisfying the same equation (see [4] or the corresponding result for rings by Jacobson [2]). Hence, by Corollary 1.2, every zerosymmetric nearring with 1 that satisfies x2m ·9+1 = x for
AUTOMORPHISM GROUPS OF ULTRAPRODUCTS OF FINITE SYMMETRIC GROUPS
"... Abstract. It is consistent that there exists a nonprincipal ultrafilter U over N such that every automorphism of the corresponding ultraproduct Q U Sym(n) is inner. 1. ..."
Abstract
 Add to MetaCart
Abstract. It is consistent that there exists a nonprincipal ultrafilter U over N such that every automorphism of the corresponding ultraproduct Q U Sym(n) is inner. 1.
Commutators in groups definable in ominimal structures
, 2010
"... We prove the definability, and actually the finiteness of the commutator width, of many commutator subgroups in groups definable in ominimal structures. It applies in particular to derived series and to lower central series of solvable groups. Along the way, we prove some generalities on groups wit ..."
Abstract
 Add to MetaCart
We prove the definability, and actually the finiteness of the commutator width, of many commutator subgroups in groups definable in ominimal structures. It applies in particular to derived series and to lower central series of solvable groups. Along the way, we prove some generalities on groups with the descending chain condition on definable subgroups and/or with a definable and additive dimension.
The Classification of the Finite Simple Groups: An Overview
 MONOGRAFÍAS DE LA REAL ACADEMIA DE CIENCIAS DE ZARAGOZA. 26: 89–104, (2004)
, 2004
"... ..."