GROUPS AND MODEL

THEORY

A conference on the interaction between Group theory and Model theory

Mulheim a. d. Ruhr, GermanyMay 30 - June 3, 2011

ABSTRACTSof all talks

ON FINITE GROUPS WITH SUBGROUPSSATISFYING A GENERALIZED EMBEDDED

PROPERTYKHALED AL-SHARO

All groups considered are finite. Recall that a subgroup H of a groupG is said to be S-permutable, S-quasinormal, or π(G)-quasinormal in G ifHP = PH for all Sylow subgroups P of G. A subgroup A of a group G issaid to be S-quasinormally embedded G or S-permutably embedded in G ifeach Sylow subgroup of A is also a Sylow of some S-permutable subgroupof G. A subgroup A is said to be weakly S-supplemented in G if for somesubgroup T of G we have AT = G and A ∩ T ≤ AsG where AsG is thesubgroup generated by all those subgroups of H which are S-permutable inG.

In this talk, we analyze the following generalization of last two concepts.LetH be a subgroup of a groupG. Then we say thatH is nearly S-permutablyembedded in G if G has a subgroup T and an S-permutably embedded sub-group C such that HT = G and T ∩H ≤ C ≤ H.

It is clear that every S-permutably embedded subgroup and every weaklyS-supplemented subgroup are nearly S-permutably embedded subgroup. Anexample will be constructed to show that in general the set of all nearly S-permutably embedded subgroups is wider than the set of all S-permutablyembedded subgroups and the set of all weakly S-supplemented subgroups.Some other related results will be considered in this talk.

K. AL-SHARO, DEPARTMENT OF MATHEMATICS, AL AL-BAYT UNIVERSITY, JORDANE-mail address: sharo kh@yahoo.com

ALMOST COMPLETELY DECOMPOSABLEGROUPS AND UNBOUNDED REPRESENTATION

TYPEDAVID M. ARNOLD

(joint work with Adolf Mader, Otto Mutzbauer and Ebru Solak)

Let p be a prime, S a finite set of p-locally free types, and m a naturalnumber. An S−group with pm-regulator quotient is an almost completelydecomposable groupG with critical typeset contained in S and pmG ⊆ R(G),

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the regulator subgroup of G. Sufficient conditions on S and m are givenfor S−groups with pm-regulator quotient to have unbounded representationtype, i.e. there are indecomposable groups of arbitrarily large finite rank.It is conjectured that these conditions are also necessary if S is an invertedforest.

D. ARNOLD, DEPARTMENT OF MATHEMATICS, BAYLOR UNIVERSITY, WACO, TX 76798,U.S.A.

E-mail address: David Arnold@baylor.edu

ON THE ABELIANIZATION OF DERIVEDCATEGORIES AND A NEGATIVE SOLUTION TO

ROSICKY’S CONJECTURESILVANA BAZZONI

(joint work with Jan Stovıcek)

We prove for a large family of rings R that their λ-pure global dimensionis greater than one for each infinite regular cardinal λ. This answers in neg-ative a problem posed by Rosicky. The derived categories of such rings thendo not satisfy the Adams λ-representability for morphisms for any λ. Equiv-alently, they are examples of well generated triangulated categories whoseλ-abelianization in the sense of Neeman is not a full functor for any λ. Inparticular we show that given a compactly generated triangulated category,one may not be able to find a Rosicky functor among the λ-abelianizationfunctors.

S. BAZZONI, DIPARTIMENTO DI MATEMATICA PURA E APPLICATA, UNIVERSITY OFPADOVA, 35121 PADOVA, ITALY

E-mail address:bazzoni@math.unipd.itHomepage: http://www.math.unipd.it/∼bazzoni

GENERALIZATIONS OF LOCAL CRQ-GROUPTHEORY

EKATERINA BLAGOVESHCHENSKAYA(joint work with Rudiger Gobel and Lutz Strungmann)

In our work a new class of torsion-free abelian groups of infinite rankadmitting non-isomorphic direct decompositions into indecomposable sum-mands is introduced. Groups of this class are defined as epimorphic imagesof some local crq-groups.

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A structure theorem describing possible direct decompositions of suchgroups has been proved and a complete classification of them has been ob-tained up to near isomorphism.

E. BLAGOVESHCHENSKAYA, DEPARTMENT OF MATHEMATICS, PETERSBURG STATETRANSPORT UNIVERSITY, 9 MOSKOVSKY PROSPECT, ST. PETERSBURG 190031, RUSSIA

E-mail address: kblag2002@yahoo.com

RANDOM ABELIAN GROUPS WITHPRESCRIBED ENDOMORPHISM RINGS

GABOR BRAUN(joint work with Sebastian Pokutta)

There is a whole bunch of constructions of abelian groups with prescribedendomorphism rings. We show that many of them can be simplified byrandomizing: instead of choosing parameters to satisfy technical conditions,it suffices to choose them randomly.

So far, we have only basic examples to demonstrate the idea but westrongly believe that this is just the tip of the iceberg: randomization canenter much more deeply, and lead to natural, intuitive constructions.

G. BRAUN, FAKULTAT FUR MATHEMATIK, UNIVERSITAT DUISBURG-ESSEN, 45117 ES-SEN, GERMANY

E-mail address: braung@renyi.huHomepage: http://www.renyi.hu/∼braung/

T.B.A.PETER CAMERON

P. CAMERON, SCHOOL OF MATHEMATICAL SCIENCES, UNIVERSITY OF LONDON, LON-DON E1 4NS, U. K.

E-mail address: P.J.Cameron@qmul.ac.ukHomepage: http://www.maths.qmul.ac.uk/∼pjc/

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DEFINABLE ORTHOGONALITY CLASSES INACCESSIBLE CATEGORIES ARE SMALL

CARLES CASACUBERTA(joint work with Joan Bagaria, A. R. D. Mathias and Jirı Rosicky)

We lower substantially the strength of the assumptions needed for thevalidity of certain results in category theory which were known to follow fromVopenka’s principle. We prove that the necessary large-cardinal hypothesesdepend on the complexity of the formulas defining the given classes, in thesense of the Levy hierarchy. For example, the statement that, for a class Sof morphisms in an accessible category C, the orthogonal class of objects S⊥is a small-orthogonality class (hence reflective, if C is cocomplete) is provablein ZFC if S is Σ1, while it follows from the existence of a proper class ofsupercompact cardinals if S is Σ2, and from the existence of a proper classof what we call C(n)-extendible cardinals if S is Σn+2 for n ≥ 1. Thesecardinals form a new hierarchy, and we show that Vopenka’s principle isequivalent to the existence of C(n)-extendible cardinals for all n.

This has relevant consequences in group theory and in homotopy theory.For instance, if L is a localization on the category of groups and S is the classof L-equivalences, then L is an f -localization for some homomorphism f ifS is Σ1, or if there is a proper class of supercompact cardinals and S is Σ2,or if there is a proper class of C(n)-extendible cardinals for some n ≥ 1 andthe class S is Σn+2.

C. CASACUBERTA, DEPARTAMENT D’ALGEBRA I GEOMETRIA, UNIVERSITAT DE BARCELONA,GRAN VIA 585, 08007 BARCELONA, SPAIN

E-mail address: carles.casacuberta@ub.eduHomepage: http://atlas.mat.ub.es/personals/casac

LOOKING FOR RIGHT BOUNDED COMPLEXESGABRIELLA D’ESTE

We will present several “combinatorial” constructions of indecomposableright bounded complexes C, with projective components, with the propertyof being “orthogonal” to a rather special tilting complexes. More generally,we will point out “discrete” properties of more or less abstract tilting objects.

G. D’ESTE, DIPARTIMENTO DI MATEMATICA, UNIVERSIT DEGLI STUDI DI MILANO, VIASALDINI 50, 20133 MILANO, ITALY

E-mail address: gabriella.deste@unimi.it

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COUNTABLE RANDOM p-GROUPS WITHPRESCRIBED ULM-INVARIANTS

MANFRED DROSTE(joint work with Rudiger Gobel)

We present a probabilistic construction of countable abelian p-groups withprescribed Ulm-sequence. Our main result provides a different proof for theexistence theorem of abelian p-groups with any given countable Ulm-sequencedue to Ulm (1933), which is sometimes called Zippin’s theorem. The basicidea, applying probabilistic arguments, comes from a result by Erdos andRenyi (1963). They gave an amazing probabilistic construction of countablegraphs which with probability 1, produces the universal homogeneous graph,therefore also called the random graph. Cameron says about this in his book“Oligomorphic Permutation Groups” (1990, pp. 86,87): In 1963, Erdos andRenyi proved the following paradoxical result. . . . It is my contention thatmathematics is unique among academic pursuits in that such an apparentlyoutrageous claim can be made completely convincing by a short argument.The algebraic tool in the present paper needs methods developed in the70th of the last century, the theory of valuated abelian p-groups. Valuatedabelian p-groups are natural generalizations of abelian p-groups with theheight valuation, investigated in detail by Richman and Walker (1973, 1979)and others. We have to establish extensions of finite valuated abelian p-groups dominated by a given Ulm-sequence f . Our probabilistic constructionproduces a countable valuated abelian p-group with Ulm sequence dominatedby f . Our main result shows that with probability 1, the constructed grouphas precisely f as its Ulm sequence.

Probabilistic results of similar nature have been obtained for Scott-domainsin theoretical computer science (with D. Kuske) and for causal sets in theo-retical physics.

M. DROSTE, INSTITUT FUR INFORMATIK, UNIVERSITAT LEIPZIG, 04009 LEIPZIG, GER-MANY

E-mail address: droste@informatik.uni-leipzig.deHomepage: http://www.informatik.uni-leipzig.de/∼droste/

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THE JACOBSON RADICAL’S ROLE INISOMORPHISM THEOREMS

MARY FLAGG

The Baer-Kaplansky theorem states that two torsion groups are isomor-phic if and only if their endomorphism rings are isomorphic rings. Let R bea discrete valuation domain. If R is complete in the p-adic topology, it is wellknown that torsion-free R-modules and several classes of mixed R-modules,including modules with finite torsion-free rank and a totally projective tor-sion submodule, also satisfy a Baer-Kaplansky type isomorphism theorem. Iwill show that in several of these classes an isomorphism between only theJacobson radical of the endomorphsim ring of two R-modules is sufficient toprove that the modules are isomorphic. Furthermore, in most cases (even ifthe ring R is not complete) the Jacobson radical of the endomorphism ringcontains sufficient information to determine the torsion submodule of an R-module.

M. FLAGG, DEPARTMENT OF MATHEMATICS, UNIVERSITY OF HOUSTON, HOUSTON,TEXAS 77204-3008, USA

E-mail address: mflagg@math.uh.edu

COHOMOLOGY IN O-MINIMAL EXPANSIONS OFGROUPS

ANTONGIULIO FORNASIERO

Let K be an o-minimal structure and X be a definable set. If K expandsa field, one can use the triangulation theorem to define the cohomology of Xas the simplicial cohomology of a triangulation of X (see A. Woerheide PhDthesis).

If K expands only a group, one instead defines the (o-minimal) coho-mology of X as the sheaf cohomology of its space of types, endowed withthe spectral topology (instead of the usual Stone topology). With A. Be-rarduccil, we proved that the cohomology groups of X (with coefficient in afinitely generated Abelian group) are finitely generated and invariant underelementary extensions and o-minimal expansions, provided that X is defin-ably compact.

A. FORNASIERO, FACHBEREICH MATHEMATIK UND INFORMATIK, UNIVERSITAT MUNSTER,48149 MUNSTER, GERMANY

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E-mail address: antongiulio.fornasiero@googlemail.comHomepage: http://www.dm.unipi.it/∼fornasiero/

WEAK-INJECTIVE MODULESLASZLO FUCHS

Weak-injective modules are defined as modulesM satisfying Ext1R((X,M))

= 0 for all modules X of weak dimension ≤ 1. They were studied overdomains by S.B. Lee and myself, and now we are investigating them overarbitrary commutative rings. Many of the pleasant properties shared in thedomain case are lost in general, but weak-injective envelopes remain closelyrelated to flat covers over any commutative ring.

L. FUCHS, MATHEMATICS DEPARTMENT, TULANE UNIVERSITY, NEW ORLEANS, LA70118, USA

E-mail address: fuchs@tulane.eduHomepage: http://www.math.tulane.edu/faculty/fuchs.html

BOUNDED SIMPLICITY OF ISOTROPIC GROUPSJAKUB GISMATULLIN

Group G is called boundedly simple if there exists a natural number Nsuch that for every two noncentral elements x, y ∈ G, element x can bewritten as a product of N conjugates of y and y−1.

Let k be an arbitrary infinite field and let G be a connected, reductive k-group which is almost k-simple and isotropic over k. In 1964 Tits proved thatthe group G(k) of k-rational points of G contains certain Zariski dense nor-mal subgroup G(k)+ with the property that every subgroup of G(k) which isnormalised by G(k)+ either is central or contains G(k)+, hence the quotientof G(k)+ by its center is a simple group as an abstract group. The structureof the group G(k)+ is not very well understood in general. The Kneser-Titsproblem asks if G(k) = G(k)+ when G is simply-connected, almost k-simpleand isotropic over k. A particular case of the Kneser-Tits problem for groupsof inner type (form) 1An has been known as the Tannaka-Artin conjecture: IsSL1(D) = [D×, D×]? In this conjecture D is a finite dimensional central divi-sion simple k-algebra, SL1(D) is the subgroup of D× of elements of reducednorm 1 and [D×, D×] is the commutator subgroup of D×. Both conjectureshas been proved false by Platonov in 1975, by considering biquaternion alge-bras over field Q(x, y).

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Looking from the model theoretical point of view we would like to knowwhether the group G(k)+ is boundedly simply or more generally whetherG(k)+ is first order definable in the structure (G(k), ·).

We say that connected reductive k-group G is W -trivial if for every fieldextension F of k, G(F ) = G(F )+. Is has been proved by Gille that k-rationalgroups areW -trivial, so satisfy the Kneser-Tits problem. Chernousov, Merkur-jev and Platonov proved that e.g. k-quasi split groups and groups of typeBn, Cn are k-rational.

We proved that in the following cases G(k)+ is boundedly simple:

(i) G is semisimple, almost k-simple and the universal k-cover G of G isW -trivial (e.g. groups of type Bn, Cn),

(ii) G is of inner type 1An i.e. G(k) is SLr(D) for r ≥ 2 and some centraldivision algebra D with finite commutator width of [D×, D×].

By results of Knus, Merkurjev, Rost and Tignol from The Book of Involu-tions, every biquaternion algebra D has finite commutator width. In partic-ular, our result imply that in Platonov counterexample G to the Kneser-Titsproblem, group G(k)+ is boundedly simple and definable.

J. GISMATULLIN, SCHOOL OF MATHEMATICS, UNIVERSITY OF LEEDS, WOODHOUSELANE, LEEDS, LS2 9JT, UK

andINSTYTUT MATEMATYCZNY UNIWERSYTETU WROCLAWSKIEGO, PL. GRUNDWALDZKI

2/4, 50-384 WROCLAW, POLANDE-mail address: gismat@math.uni.wroc.pl, www.math.uni.wroc.pl/˜gismat

A FINITELY PRESENTED ORDERABLE GROUPWITH INSOLUBLE WORD PROBLEM

ANDREW GLASS(joint work with Vasily Bludov)

A. GLASS, CENTRE FOR MATHEMATICAL SCIENCES, UNIVERSITY OF CAMBRIDGE, WILBER-FORCE ROAD, CAMBRIDGE CB3 0WA, U.K.

E-mail address: A.M.W.Glass@dpmms.cam.ac.ukHomepage: http://www.dpmms.cam.ac.uk/∼amwg/

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OLD RESULTS AND NEW THEOREMS ONABSOLUTE CONSTRUCTIONS

RUDIGER GOBEL(joint work with Sebastian Pokutta)

The notion of absolute formulas in set theory can be rephrased andslightly extended by saying that a property of an algebraic object holds ab-solutely if it remains the same the same in every generic extension of the (settheoretic) universe. Hence these objects are not effected by forcing argument.And if we insist on absolute constructions, then many familiar methods con-structing these objects (in particular those involving stationary sets, like theShelah elevator and Black Box predictions) must be discarded. The newconstructions require new, more refined steps which are robust under thoseearthquakes of the universe by forcing. The constructions are more ‘finitary’,which is another advantage of the requested changes.

This approach entered into algebra through a paper by Eklof-Shelahabout 10 years ago and came into the focus of investigations since. It turnsout that objects of huge size are in most cases not permitted. There is a(sometimes precise) cardinal-barrier κ(ω) for these problems, which is thefirst ω-Erdos cardinal. In an older, but fundamental paper Shelah con-structed already absolutely rigid families of colored trees of any size λ < κ(ω).All construction of absolutely rigid families of modules, modules with distin-guished submodules, endomorphism monoids of graphs, E-rings, E-modules,rely directly or indirectly on the idea to encode Shelah’s trees into these newobjects to get absolutely rigid families (in the most obvious sense). Oftenthe indirect approach is more convenient, and we embed absolutely rigid R5-modules (modules with five distinguished submodules, already obtained bytree-embeddings) into the structure we are interested in.

I will give a survey on these results and explain more details (with thehelp of a ‘non-absolute paper’) about the construction of absolutely rigidfields with prescribed subfields K and prescribed automorphism groups byembedding into them absolutely rigid P5-modules over the prime field P ofK.

R. GOBEL, FAKULTAT FUR MATHEMATIK, UNIVERSITAT DUISBURG-ESSEN, 45117 ES-SEN, GERMANY

E-mail address: ruediger.goebel@uni-due.deHomepage: http://www.uni-due.de/algebra-logic/goebel.shtml

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ON PROJECTION-INVARIANT SUBGROUPS OFABELIAN p-GROUPS

BRENDAN GOLDSMITH(joint work with Peter V. Danchev)

A subgroup P of an Abelian p-group G is said to be projection-invariantin G, if Pπ ≤ P for all idempotent endomorphisms π in End(G). Clearlyfully invariant subgroups are projection invariant, but the converse is nottrue in general. Hausen and Megibben have, however, shown that in manyfamiliar situations these two concepts coincide. In a different direction, theauthors have previously introduced the notions of socle-regular and stronglysocle-regular groups by focussing on the socles of fully invariant and charac-teristic subgroups of p-groups. In the present work the authors examine thesocles of projection-invariant subgroups of Abelian p-groups.

B. GOLDSMITH, SCHOOL OF MATHEMATICAL SCIENCES, DUBLIN INSTITUTE OF TECH-NOLOGY, DUBLIN 8, IRELAND

E-mail address: brendan.goldsmith@dit.ie

ENDOMORPHISMS OF ABELIAN GROUPS WITHSMALL ALGEBRAIC ENTROPY

KETAO GONG(joint work with D. Dikranjan and P. Zanardo)

We investigate the endomorphism φ of Abelian group having “small” al-gebraic entropy(here small usually means its value is less than log 2) By usingessentially elementary tools from linear algebra and matrix theory, we showthat this study can be carried out in the group Qm; In such case the auto-morphism φ with small algebraic entropy must have all eigenvalues in theopen disc of radius 2, centered at 0 and φ must leave invariant a lattice inQm, i.e., be essentially an automorphism of Zm. In particular, all eigenvaluesof an automorphism φ with zero entropy must be roots of unity. This is aparticular case of a more general fact known as Algebraic Yuzvinskii Theo-rem.

K. GONG, SCHOOL OF MATHEMATICAL SCIENCES, DUBLIN INSTITUTE OF TECHNOL-OGY, DUBLIN 8, IRELAND

E-mail address: ketao.gong@student.dit.ie

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ℵn-FREE ENDO-RIGID GROUPS IN ZFCDANIEL HERDEN

(joint work with Rudiger Gobel and Saharon Shelah)

A group G is called endo-rigid if EndG = Z. Until now the constructionof endo-rigid groups in ZFC has been limited to groups of freeness degree atmost ℵ1-free. In our talk we want to demonstrate for every positive integern < ω the existence of an ℵn-free endo-rigid group in ZFC of cardinalityi+

2n−1. This ground breaking construction introduces new powerful combina-torial tools which will easily be adaptable to arbitrary realizations of ℵn-freegroups in ZFC.

D. HERDEN, FAKULTAT FUR MATHEMATIK, UNIVERSITAT DUISBURG-ESSEN, 45117 ES-SEN, GERMANY

E-mail address: daniel.herden@uni-due.deHomepage: http://www.uni-due.de/algebra-logic/herden.shtml

THE UNDEFINABILITY OF WARFIELDMODULES IN L∞ω

CAROL JACOBY

Ulm’s Theorem presents invariants that classify countable torsion abeliangroups up to isomorphism. Barwise and Eklof extended this result to theclassification of arbitrary torsion abelian groups up to L∞ω-equivalence. Theyproved that in L∞ω the invariants are expressible and the class of groupsis definable. The problem addressed here is a similar extension to L∞ω-equivalence of the results of Warfield classifying a certain class of mixedZp-modules up to isomorphism.

Zp-modules with partial decomposition bases have been completely clas-sified up to equivalence in L∞ω with invariants that are expressible in L∞ω.This class of modules includes the Warfield modules. We prove that thisclass is not definable in L∞ω, nor is any class that generalizes the Warfieldmodules in any reasonable way.

C. JACOBY, JACOBY CONSULTING, LONG BEACH, CALIFORNIAE-mail address: cjacoby@jacobyconsulting.com

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ON WEAKLY c-NORMAL SUBGROUPS OFFINITE GROUPS

JEHAD J. JARADEN

Throughout this work, all groups are finite. Recall that a formation is a ho-momorph F of groups such that each group G has a smallest normal subgroup(denoted by GF ) whose quotient is still in F . A formation F is said to be satu-rated if it contains each group G with G/Φ(G) ∈ F . In this worke, we use U todenote the formation of the supersoluble groups. A chief factor H/K of a group Gis called central if G = CG(H/K)). The symbol Z∞(G) denotes the hypercenter ofa group G that is the product of all such normal subgroups H of G whose G-chieffactors are central.

We say that a subgroup H of a group G is c-normal in G if there exists anormal in G subgroup T such that G = HT and T ∩H ≤ HG , where HG is thelargest normal subgroup of G contained in H.

Several authors have investigated the structure of a group G under the assump-tion that the maximal or the minimal subgroups of the Sylow subgroups of somesubgroups of G are well situated in G. Buckley proved that a group of odd orderis supersoluble if its minimal subgroups are normal. Later on, Srinivasan showedthat a group G is supersoluble if it has a normal subgroup N with supersolublequotient G/N such that all maximal subgroups of the Sylow subgroups of N arenormal in G. Ramadan proved : If G is a soluble group and all maximal subgroupsof any Sylow subgroup of F (G) are normal in G, then G is supersoluble. Wang gen-eralized the results replacing normality by c-normality. Li and Guo obtained thesame results by assuming the c-normality of the maximal or minimal subgroupsof the Fitting subgroup of a soluble group. By using the theory of formations,Wei extended further the results to a saturated formation containing the class ofsupersoluble groups U .

Definition . Let G be a group. Then we say that a subgroup bv H of Gis weakly c-normal in G if G has a normal subgroup T such that HT = G and(H ∩ T )HG/HG is contained in the hypercenter Z∞(G/HG) of G/HG.

It is clear that every c-normal subgroup of G is weakly c-normal in G. Thefollowing example shows that the inverse statement is not true in general.

Example . Let m > 3 and P = Mm(2) = 〈x, y|x2m−1= y2 = 1, xy =

x1+2m−2〉. Let H = 〈y〉. There is a 2-group G such that P ≤ G′ . Hence B ≤ Φ(G).Therefore G is the only supplement of H in G. But H is not normal in G. HenceH is not c-normal in G. It is also clear that H is weakly c-normal in G.

Our main goal here is to prove the following:Theorem 1. Let F be a saturated formation containing all supersoluble

groups and G a group with a normal subgroup E such that G/E ∈ F . Suppose

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that every non-cyclic Sylow subgroup P of F ∗(E) has a subgroup D such that1 < |D| < |P | and all subgroups H of P with order |H| = |D| and with order 2|D|(if P is a non-abelian 2-group and |P : D| > 2) are weakly c-normal in G in G.Then G ∈ F .

Theorem 2. Let F be a saturated formation containing all supersolublegroups and G a group with a normal subgroup E such that G/E ∈ F . Sup-pose that every non-cyclic Sylow subgroup P of E has a subgroup D such that1 < |D| < |P | and all subgroups H of P with order |H| = |D| and with order 2|D|(if P is a non-abelian 2-group and |P : D| > 2) are weakly c-normal in G. ThenG ∈ F .

J. J. JARADEN, DEPARTMENT OF MATHEMATICS AND STATISTICS, AL-HUSSEIN BINTALAL UNIVERSITY, MA’AN, JORDAN

E-mail address: jjaraden@mtu.eduHomepage: http://jehad.tech.officelive.com/default.aspx

RESULTS IN NIP THEORIESITAY KAPLAN

This talk will be concerned with abstract model theory. Specifically inDependent or NIP theories.This is a branch in classification theory where we try to research propertiesof first order theories which on the one hand are general enough to containinteresting examples, and on the other hand give rise to non trivial theorems.NIP theories contain stable theories but also contain some valued fields, o-minimal theories, trees and more. There is a phenomenon in which behaviorin dependent theories can be analyzed in 2 parts: the order part and thestable part (for example, this takes place in Shelah’s type decompositiontheorem).

So, generally speaking, one of the approaches to studying NIP theories isto generalize common results from the stable case and the o-minimal case.

I will present some new developments in this area.

I. KAPLAN, FACHBEREICH MATHEMATIK UND STATISTIK, UNIVERSITAT KONSTANZ,78457 KONSTANZ, GERMANY

E-mail address: deyo@math.huji.ac.il

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C-DECOMPOSABLE PRIMARY ABELIANGROUPS AND WEAK pω·2-PROJECTIVES

PATRICK W. KEEF

All groups considered will be abelian p-groups for some fixed prime p. Thegroup G is C-decomposable if G ∼= H ⊕C, where C is a direct sum of cyclicgroups and fr r(C) = fr r(G). A new characterization of C-decomposability isgiven. It implies that most reduced groups whose final ranks have countablecofinality are C-decomposable; and that it is undecidable in ZFC whether allsuch groups are C-decomposable. It is known that there are pω+2-projectivegroups that are not C-decomposable, but that for n < ω, all pω+n-projectivegroups satisfy a slightly weaker condition, i.e., they are “far from thick”(where G is pω+n-projective iff it has a subgroup P ⊆ G[pn] such that G/P isa direct sum of cyclics). This result is generalized by showing that all weakpω·2-projectives are far from thick (where G is a weak pω·2-projective iff it hasa subgroup A such that A and G/A are direct sums of cyclics). This implies,for example, that any weak pω·2-projective group G satisfies the “generalizedcore class property”; i.e., for all n < ω, G is not pω+n-projective iff it has asubgroup H that is pω+n+1-projective but not pω+n-projective.

P. KEEF, DEPARTMENT OF MATHEMATICS, WHITMAN COLLEGE, WALLA WALLA, WA99362, USA

E-mail address: keef@whitman.edu

THE ISOMORPHISM PROBLEM FORAUTOMATIC STRUCTURES

DIETRICH KUSKE(joint work with Jimaou Liu and Markus Lohrey)

Automatic structures are recursive structures where the universe and allrelations can be recognized by finite automata. Differently from general re-cursive structure, every automatic structure has a decidable first-order theorywhich is the main motivation for their investigation. Since this decidabilityholds uniformly, the elementary equivalence problem for automatic structuresis in Π0

1 (the first universal level of the arithmetical hierarchy). It is knownthat the isomorphism problem for automatic structures is complete for Σ1

1

(the first existential level of the analytical hierarchy). This talk discussesseveral new results on isomorphism problems for automatic structures:

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(i) The isomorphism problem and the elementary equivalence problem forautomatic equivalence relations are complete for Π0

1.

(ii) The isomorphism problem for automatic linear orders and for automatictrees are complete for Σ1

1.

These results solve some open questions of Khoussainov, Rubin, and Stephan.

D. KUSKE, FG THEORETISCHE INFORMATIK, TU ILMENAU, POB 100565, 98684 ILMENAU,GERMANY

E-mail address: dietrich.kuske@tu-ilmenau.deHomepage: http://eiche.theoinf.tu-ilmenau.de/kuske/

INFINITARY EQUIVALENCE OF ABELIANGROUPS WITH PARTIAL DECOMPOSITION

BASESPETER LOTH

(joint work with Carol Jacoby, Katrin Leistner and Lutz Strungmann)

We consider the class of abelian groups possessing partial decompositionbases in the infinitary language Lδ∞ω for δ an ordinal. This class containsthe class of Warfield groups which are direct summands of simply presentedgroups or, alternatively, are abelian groups possessing a nice decompositionbasis with simply presented cokernel. In this paper, we give a classificationtheorem using numerical invariants that are deduced from the classical Ulm-Kaplansky and Warfield invariants. This extends earlier work by Barwise-Eklof, Gobel and the authors of this joint work.

P. LOTH, DEPARTMENT OF MATHEMATICS, SACRED HEART UNIVERSITY, FAIRFIELD,CT 06825, USA

E-mail address: LothP@sacredheart.eduHomepage: http://faculty.sacredheart.edu/lothp

THE HEIGHT OF THE AUTOMORPHISM TOWEROF A CENTERLESS GROUP

PHILIPP LUCKE

Let G be a centreless group. We call a sequence 〈Gα | α ∈ On〉 of groupsan automorphism tower of G if the following statements hold.

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(i) G = G0.

(ii) For all α ∈ On, Gα is a normal subgroup of Gα+1 and the induced map

ϕα : Gα+1 −→ Aut(Gα); h 7−→[g 7→ gh

]is an isomorphism.

(iii) If λ is a limit ordinal, then Gλ =⋃α<λGα.

It is easy to see that each group Gα is uniquely determined up to isomor-phisms that induce the identity map on G.

Simon Thomas showed that for every infinite centreless group G of car-dinality κ, there is an α < (2κ)+ with Gα = Gα+1 and therefore Gα = Gβ

for all β ≥ α. We let τ(G) denote the minimal ordinal with this property.Given an infinite cardinal κ, we define

τκ = lub{τ(G) | G is a centreless group of cardinality κ}.

By Thomas’ result, τκ < (2κ)+ holds. The following problem is still open.

Problem. Find a model 〈M,∈M〉 of ZFC and an infinite cardinal κ in Msuch that it is possible to “compute” the exact value of τκ in M .

Building upon results and methods developed by Itay Kaplan and Sa-haron Shelah, I want to show how smaller upper bounds for τκ can be ob-tained by combining group-theoretic arguments with admissible set theoryand fine structure theory for the inner model L(P(κ)).

P. LUCKE, FACHBEREICH MATHEMATIK UND INFORMATIK, UNIVERSITAT MUNSTER,EINSTEINSTRASSE 62, 48149 MUNSTER, GERMANY

E-mail address: philipp.luecke@uni-muenster.de

MODEL THEORY OF PSEUDOFINITE GROUPSDUGALD MACPHERSON

A group is pseudofinite if it is an infinite model of the theory of finitegroups. I will discuss connections between pseudofiniteness (for groups) andgeneralisations of model theoretic stability theory. The particular focus willbe on pseudofinite groups (and permutation groups) whose first order theory

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is supersimple. This includes groups of Lie type over pseudofinite fields, andsome model theory was developed in recent joint work with Elwes, Jaligot,and Ryten.

H. D. MACPHERSON, DEPARTMENT OF PURE MATHEMATICS, UNIVERSITY OF LEEDS,LEEDS, LS2 9JT, U.K.

E-mail address: H.D.MacPherson@leeds.ac.ukHomepage: http://www.amsta.leeds.ac.uk/pure/staff/macpherson/macpherson.html

GROUPS WHERE FREE SUBGROUPS AREABUNDANT

ZACHARY MESYAN

Given an infinite topological group G and a cardinal κ > 0, we say that Gis almost κ-free if the set of κ-tuples (gi)i∈κ ∈ Gκ which freely generate freesubgroups of G is dense in Gκ. I will discuss groups having this property,which generalizes a notion introduced by Gartside and Knight. The initialinspiration for studying such groups came from a 1990 result of Dixon, show-ing that the group Sym(N) of all permutations of the natural numbers isalmost n-free for all integers n ≥ 2. It turns out that there is a number ofother groups with this property. For instance, given a cardinal κ > 0, anynon-discrete Hausdorff topological group that contains a dense free subgroupof rank κ is almost κ-free.

Z. MESYAN, DEPARTMENT OF MATHEMATICS, UNIVERSITY OF COLORADO, COLORADOSPRINGS, CO 80933, U.S.A.

E-mail address: zmesyan@uccs.eduHomepage: http://www.uccs.edu/∼zmesyan/

A REVISITATION OF BUTLER’S THEOREMCLAUDIA METELLI

Call c.d.-group a completely decomposable Abelian group of finite rank.Butler’s Theorem proves that the class of pure subgroups of c.d. groupsequals the class of torsionfree quotients of c.d.-groups. While one implicationof the equivalence is proved by an applicable construction, the other is a farfrom applicable finite induction, and even its later proof in Arnold’s book isnot constructive.

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We give an alternative proof of this implication by providing a workablealgorithm, which minimizes the construction in certain cases, building notonly the c.d. container of the Butler group but also the inclusion morphism.

C. METELLI, DIPARTIMENTO DI MATEMATICA E APPLICAZIONI, UNIVERSITA’ FED-ERICO II DI NAPOLI, ITALY

E-mail address: cmetelli@math.unipd.it

GENERATING THE INFINITE SYMMETRICGROUP USING A CLOSED SUBGROUP AND THE

LEAST NUMBER OF OTHER ELEMENTSJAMES D. MITCHELL

Let G be a group or semigroup, and let H be a fixed subgroup or subsemi-group of G. In this talk I will discuss the problem of determining the leastcardinality of U ⊆ G such that 〈: H,U : 〉 = G. In particular, if G := S∞is the infinite symmetric group and H is closed in the usual Polish topology,then using theorems of Galvin, and Bergman and Shelah we can prove that|U | ∈ {0, 1, d, c} where d denotes the so-called dominating number, and c

denotes the cardinality of the continuum.

J. D. MITCHELL, MATHEMATICAL INSTITUTE, NORTH HAUGH, ST. ANDREWS, FIFE,KY16 9SS, SCOTLAND

E-mail address: jdm3@st-and.ac.ukHomepage: http://www-groups.mcs.st-andrews.ac.uk/∼jamesm/

GEOMETRY OF ORTHOGONAL INVOLUTIONSMARK PANKOV

M. PANKOV, FACULTY OF MATHEMATICS AND COMPUTER SCIENCE, UNIVERSITY OFWARMIA AND MAZURY, 10-561 OLSZTYN, POLAND

E-mail address: pankov@matman.uwm.edu.plHomepage: http://wmii.uwm.edu.pl/∼pankov/

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ON κ-ARTINIAN AND κ-NOETHERIAN RINGSKULUMANI M. RANGASWAMY

Let κ be an infinite cardinal. A ring R is called a right κ-artinian if forevery descending chain of right ideals

I1 ⊇ · · · ⊇ Iα ⊇ Iα+1 ⊇ · · · (α < λ),

the set S of ordinals 1 ≤ α < λ such that Iα 6= Iα+1 has cardinality less thanκ. A κ-noetherian ring is anologously defined. We derive several properties ofκ-artinian rings R including the annihilator conditions on R-modules. Simi-larly, we obtain characterizing properties of κ-noetherian rings R in terms ofinjective modules.

K. RANGASWAMY, DEPARTMENT OF MATHEMATICS, UNIVERSITY OF COLORADO, COL-ORADO SPRINGS, CO 80918, USA

E-mail address: ranga@uccs.eduHomepage: http://www.uccs.edu/∼krangasw/

SOME RESULTS AND OPEN QUESTIONS ONENVELOPES AND COVERS FOR GROUPS

JOSE L. RODRIGUEZ(joint work with S. Estrada)

In this talk we consider envelopes and covers of groups, and their associ-ated localizations and cellular covers. Our aim is to encounter recent workby Enochs, Rada and Hill in module approximation theory with work under-taken by abelian group theorists and algebraic topologists in the context ofhomotopical localization and cellularization of spaces.

J. L. RODRIGUEZ, AREA DE GEOMETRIA Y TOPOLOGIA, FACULTAD DE CIENCIAS EX-PERIMENTALES, UNIVERSITY OF ALMERIA, LA CANADA DE SAN URBANO, 04120 ALMERIA,SPAIN

E-mail address: jlrodri@ual.es

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THE UNIVERSAL PROPERTY OF THEALGEBRAIC ENTROPY

LUIGI SALCE(joint work with Peter Vamos and Simone Virili)

Let entL denote the algebraic entropy associated to a discrete lengthfunction L : Mod(R) → R ≥ 0 ∪ {∞} from the category of left modulesover a ring R to the non-negative real numbers plus ∞. The UniversalProperty of the algebraic entropy entL says that, passing to modules overthe polynomial ring R[X], there exists a unique discrete length functionLX : lF inL(R[X]) → R ≥ 0 ∪ {∞} from the subcategory lF inL(R[X]) ofMod(R[X]) consisting of the locally finite modules, which extends L in atechnical sense, provided that LX(N) = 0 for R[X]-modules N such thatL(N) < ∞. The length function LX coincides with the restriction of thealgebraic entropy entL to lF inL(R[X]).

L. SALCE, DIPARTIMENTO DI MATEMATICA PURA E APPLICATA, UNIVERSITY OF PADOVA,35121 PADOVA, ITALY

E-mail address: salce@math.unipd.itHomepage: http://www.math.unipd.it/∼salce/

FUNCTORIAL PROPERTIES OF HOM AND EXTPHILL SCHULTZ

(joint work with Simion Breaz)

Let F be a covariant functor on a category of modules which shares withHom(G,−) the property of preserving products, i.e., F(

∏iAi) is naturally

isomorphic to∏

iF(Ai). When does F preserve direct sums from some classC of modules, map direct sums from C to direct products, or map directproducts from C to direct sums? It turns out that only the first of theseproblems is non–trivial, corresponding to the case of Hom(G,−) when Gis a small or self–small abelian group. We apply these general results toHom(G,−) and Ext(G,−) using various classes C of abelian groups.

The results can be dualised: let F be a contravariant functor which,like Hom(−, G), inverts direct sums, i.e., F(⊕iAi) is naturally isomorphic to∏

iF(Ai). When does F preserve direct sums or products, or invert directproducts from some class C? Once again, of all possible preserving or invert-ing properties of F , only that of inverting direct products is non–trivial, cor-responding to the case of Hom(−, G) when G is a slender abelian group. The

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results are again applied to Hom(−, G) and Ext(−, G) with several classesC.

P. SCHULTZ, SCHOOL OF MATHEMATICS AND STATISTICS, THE UNIVERSITY OF WEST-ERN AUSTRALIA, NEDLANDS, 6009, AUSTRALIA

E-mail address: schultz@maths.uwa.edu.au

EXISTENTIALLY CLOSED LOCALLY FINITEGROUPS

SAHARON SHELAH

We investigate this class of groups originally called ulf (universal locallyfinite groups) of cardinality λ. We prove that for every locally finite groupG there is a canonical existentially closed extention of the same cardinality,unique up to isomorphism and increasing with G. Also we get, e.g. existenceof complete members (i.e. with no non-inner automorphisms) in many car-dinals extending a given member; provably in ZFC. We also get a parallel tostability theory in the sense of investigating definable types. This is basedon the author’s paper [312], wich exists in the mathematical arXive and con-tinuations which are in preparation.

S. SHELAH, EINSTEIN INSTITUTE OF MATHEMATICS, EDMOND J. SAFRA CAMPUS, GI-VAT RAM, THE HEBREW UNIVERSITY OF JERUSALEM, JERUSALEM, 91904, ISRAEL andDEPARTMENT OF MATHEMATICS, HILL CENTER - BUSCH CAMPUS, RUTGERS - THE STATEUNIVERSITY OF NEW JERSEY, 110 FRELINGHUYSEN ROAD, PISCATAWAY, NJ 08854-8019, USA

E-mail address: shelah@huji.ac.ilHomepage: http://shelah.logic.at

INDECOMPOSABLE (1, 3)-GROUPSEBRU SOLAK

(joint work with David M. Arnold, Adolf Mader and Otto Mutzbauer)

A torsion free abelian group of finite rank is called almost completely de-composable if it has a completely decomposable subgroup of finite index. A(1, 3)-group G is a p-local, p-reduced almost completely decomposable groupwith critical typeset Tcr(G) = {τ0, τ1, τ2, τ3} where τ0 is incomparable withτ1, τ2, τ3 and τ1 < τ2 < τ3. A p-reduced almost completely decomposablegroup G with regulator quotient a finite p-group is associated to an integercoordinate matrix. It is possible to determine the near-isomorphism classes

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of indecomposable (1, 3)-groups by using coordinate matrices. There are,depending on the exponent of the regulator quotient pk, either no indecom-posable (1, 3)-groups if k ≤ 2; only six near isomorphism classes of indecom-posable (1, 3)-groups if k = 3; and indecomposable (1, 3)-groups of arbitrarylarge rank if k ≥ 4.

E. SOLAK, DEPARTMENT OF MATHEMATICS, MIDDLE EAST TECHNICAL UNIVERSITY,06531 ANKARA, TURKEY

E-mail address: esolak@metu.edu.trHomepage: http://www.metu.edu.tr/∼esolak/

THE CLASSIFICATION PROBLEM FOR S-LOCALTORSION-FREE ABELIAN GROUPS OF FINITE

RANKSIMON THOMAS

If S is a set of primes, then an abelian group A is said to be S-local ifA is p-divisible for all primes p /∈ S. In this talk, I will prove that if n ≥ 2and S, T are sets of primes, then the classification problem for the S-localtorsion-free abelian groups of rank n is Borel reducible to the classificationproblem for the T -local torsion-free abelian groups of rank n if and only ifS ⊆ T .

S. THOMAS, MATHEMATICS DEPARTMENT, RUTGERS UNIVERSITY, 110 FRELINGHUY-SEN ROAD, PISCATAWAY, NJ 08854-8019, USA

E-mail address: sthomas@math.rutgers.eduHomepage: http://www.math.rutgers.edu/∼sthomas/

TILTING FOR COMMUTATIVE NOETHERIANRINGS

JAN TRLIFAJ

Tilting over commutative rings has for a long time been considered triv-ial because of the fact that in this case, there are no non-projective finitelygenerated tilting modules. However, the picture changes completely once weadmit infinitely generated modules in the scene. Not only that approximationtheory and category equivalences survive in this generality, but a completeclassification is possible in many cases. In this talk, we will present the rele-vant general results, and then deal with the case of commutative noetherian

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rings in particular.

J. TRLIFAJ, FACULTY OF MATHEMATICS AND PHYSICS, CHARLES UNIVERSITY, 186 75PRAGUE 8, CZECH REPUBLIC

E-mail address: trlifaj@karlin.mff.cuni.czHomepage: http://www.karlin.mff.cuni.cz/∼trlifaj/en/

GENERIC ENDOMORPHISMS OFHOMOGENEOUS STRUCTURES

JOHN K. TRUSS

I define a notion of ‘genericity’ for endomorphisms, extending an earlierdefinition which applied just to automorphisms. There are 6 natural monoidsassociated with any structure (usually assumed to be countable and to pos-sess some form of homogeneity), which are the automorphisms, embeddings,endomorphisms, monomorphisms, epimorphisms, and bimorphisms. Thereis a notion of ‘generic’ associated with each of these. I shall define thesemonoids and give some examples and results about generics in all of thesecases.

J. K. TRUSS, DEPARTMENT OF PURE MATHEMATICS, UNIVERSITY OF LEEDS, LEEDS,LS2 9JT, U.K.

E-mail address: pmtjkt@leeds.ac.ukHomepage: http://www.amsta.leeds.ac.uk/pure/staff/truss/truss.html

THE BLOCK SUM MONOID OF A RINGPETER VAMOS

P. VAMOS, SCHOOL OF MATHEMATICAL SCIENCES, UNIVERSITY OF EXETER, EXETER,EX4 4QE, U. K.

E-mail address: p.vamos@ex.ac.ukHomepage: http://empslocal.ex.ac.uk/people/staff/PVamos/

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ON MAXIMAL RELATIVELY DIVISIBLESUBMODULES

PAOLO ZANARDO(joint work with Brendan Goldsmith)

A torsion-free moduleM over an integral domain R has relatively divisible(RD-) submodules which are maximal with respect to inclusion. There aresituations in which the number of non-isomorphic maximal RD-submodulesis small; Gobel and Goldsmith have investigated this and related questionsin the context of Abelian groups. We address corresponding problems formodules over arbitrary domains. We obtain results relating to the levelof coherency of a ring R, and establish connections between the level of co-herency and the minimum number of generators of RD-submodules of a givenR-module. Under some natural restrictions, we prove that an R-module G,all of whose maximal RD-submodules are isomorphic to a fixed free moduleX of infinite rank, is itself free. We investigate R-modules G all of whosemaximal RD-submodules are isomorphic toRλ, where |R| and λ are infinitecardinals which are not too large. We first show that, for any slender integraldomain R, the module Rλ has infinitely many non-isomorphic maximal RD-submodules.Moreover, when R is a slender valuation domain with p.d.Q = 1,and G is an R-module with all maximal RD-submodules isomorphic to Rλ,we prove that G itself is isomorphic to Rλ. Consequently, in a wide rangeof situations no such module can exist, for instance if R is either a maximalPrufer domain or a DVR.

P. ZANARDO, DIPARTIMENTO DI MATEMATICA, VIA TRIESTE 63, 35121 PADOVA, ITALYE-mail address: paolo.zanardo@unipd.it

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