Splitting Mixed Groups of Finite Torsion-Free Rank

Abstract

First, we give a necessary and sufficient condition for torsion-free finite rank subgroups of arbitrary abelian groups to be purifiable. An abelian group G is said to be a strongly ADE decomposable group if there exists a purifiable T(G)-high subgroup of G. We use a previous result to characterize ADE decomposable groups of finite torsion-free rank. Finally, in an extreme case of strongly ADE decomposable groups, we give a necessary and sufficient condition for abelian groups of finite torsion-free rank to be splitting.     

STACKED BASES FOR HOMOGENEOUS COMPLETELY DECOMPOSABLE GROUPS

Generalizing a theorem by P. Hill and C. Megibben, fixing a rational group R, we characterize by numerical invariants R-presentations of a group G, namely, short exact sequences of the form 0 → A → X → G → 0, where A and X are homogeneous completely decomposable groups of the same type R. This characterization sets afloat the class of the “uniquely R-presented groups”. This class is investigated in connection with the extension to arbitrary groups of the Warfield equivalence between categories of torsionfree abelian groups induced by the functors Hom(R, –) and R ? ?. As an application, the stacked bases theorem proved by J. Cohen and H. Gluck in 1970 is extended to arbitrary pairs of homogeneous completely decomposable abelian groups of the same type.

     

On Purity and Quasiequality of Abelian Groups

We study the Cohn purity in an abelian group regarded as a left module over its endomorphism ring. We prove that if a finite rank torsion-free abelian group G is quasiequal to a direct sum in which all summands are purely simple modules over their endomorphism rings then the module _E(G) G is purely semisimple. This theorem makes it possible to construct abelian groups of any finite rank which are purely semisimple over their endomorphism rings and it reduces the problem of endopure semisimplicity of abelian groups to the same problem in the class of strongly indecomposable abelian groups.     

SPECTRALITY FOR MATRICES OF FOURIER MULTIPLIER OPERATORS ACTING IN Lp -SPACES OVER LCA GROUPS

Abstract

N. Dunford and J.T. Schwartz gave a complete characterization of those matrices of (bounded) Fourier multiplier operators acting in L ²(R^N) ⁿ which are spectral operators, [4; ch. XV]. In the present note this characterization is extended to the setting of L^P(G)ⁿ , where G is a locally compact abelian group and 1 < p < ∞; see Theorem 2.     

Phylogenetic Diversity Over an Abelian Group

There is a natural way to associate to any tree T with leaf set X, and with edges weighted by elements from an abelian group G, a map from the power set of X into G—simply add the elements on the edges that connect the leaves in that subset. This map has been well-studied in the case where G has no elements of order 2 (particularly when G is the additive group of real numbers) and, for this setting, subsets of leaves of size two play a crucial role. However, the existence and uniqueness results in that setting do not extend to arbitrary abelian groups. We study this more general problem here, and by working instead with both, pairs and triples of leaves, we obtain analogous existence and uniqueness results. Some particular results for elementary abelian 2-groups are also described. Received July 13, 2005     

The group PSL(2,q) is not the multiplication group of a loop

ABSTRACT

We introduce the notion of weak transitivity for torsion-free abelian groups. A torsion-free abelian group G is called weakly transitive if for any pair of elements x, y ∈ G and endomorphisms ?, ψ ∈ End(G) such that x? = y, yψ = x, there exists an automorphism of G mapping x onto y. It is shown that every suitable ring can be realized as the endomorphism ring of a weakly transitive torsion-free abelian group, and we characterize up to a number-theoretical property the separable weakly transitive torsion-free abelian groups.     

Monochrome symmetric subsets of colored groups

In (Electron. J. Combin. 10 (2003); http://www.combinatorics.org/volume-10/Abstracts/v1oi1r28.html), the first author (Yuliya Gryshko) asked three questions. Is it true that every infinite group admitting a 2-coloring without infinite monochromatic symmetric subsets is either almost cyclic (i.e., have a finite index subgroup which is cyclic infinite) or countable locally finite? Does every infinite group G include a monochromatic symmetric subset of any cardinal <|G| for any finite coloring? Does every uncountable group G such that |B(G)|< |G| where B(G)={xG:x²=1}, admit a 2-coloring without monochromatic symmetric subsets of cardinality |G|? We answer the first question positively. Assuming the generalized continuum hypothesis (GCH), we give a positive answer to the second question in the abelian case. Finally, we build a counter-example for the third question and we give a necessary and sufficient condition for an infinite group G to admit 2-coloring without monochromatic symmetric subsets of cardinality |G|. This generalizes some results of Protasov on infinite abelian groups (Mat. Zametki 59 (1996) 468–471; Dopovidi NAN Ukrain 1 (1999) 54–57).     

ON QUOTIENTS OF BRAID GROUPS

ABSTRACT.

Let G be the group ?[t, t ^?1] x ?. By studying the action of the braid group B_n on the set Gⁿ , we obtain representations of B_n into a wreath product of the symmetric group and the general linear group over ?[t, t ^?1]. This in particular recovers the Burau representation of the braid group. Furthermore, some quotients of the braid group are obtained by using the representations found.     

Solvable absolute Galois groups are metabelian

We show that solvable absolute Galois groups have an abelian normal subgroup such that the quotient is the direct product of two finite cyclic and a torsion-free procyclic group. In particular, solvable absolute Galois groups are metabelian. Moreover, any field with solvable absolute Galois group G admits a non-trivial henselian valuation, unless each Sylow-subgroup of G is either procyclic or isomorphic to Z ₂⋊Z/2Z. A complete classification of solvable absolute Galois groups (up to isomorphism) is given. Oblatum 22-IV-1998 & 1-IX-2000?Published online: 30 October 2000     

Locally minimal topological groups 1

The aim of this paper is to go deeper into the study of local minimality and its connection to some naturally related properties. A Hausdorff topological group (G,τ) is called locally minimal if there exists a neighborhood U of 0 in τ such that U fails to be a neighborhood of zero in any Hausdorff group topology on G which is strictly coarser than τ. Examples of locally minimal groups are all subgroups of Banach-Lie groups, all locally compact groups and all minimal groups. Motivated by the fact that locally compact NSS groups are Lie groups, we study the connection between local minimality and the NSS property, establishing that under certain conditions, locally minimal NSS groups are metrizable. A symmetric subset of an abelian group containing zero is said to be a GTG set if it generates a group topology in an analogous way as convex and symmetric subsets are unit balls for pseudonorms on a vector space. We consider topological groups which have a neighborhood basis at zero consisting of GTG sets. Examples of these locally GTG groups are: locally pseudoconvex spaces, groups uniformly free from small subgroups (UFSS groups) and locally compact abelian groups. The precise relation between these classes of groups is obtained: a topological abelian group is UFSS if and only if it is locally minimal, locally GTG and NSS. We develop a universal construction of GTG sets in arbitrary non-discrete metric abelian groups, that generates a strictly finer non-discrete UFSS topology and we characterize the metrizable abelian groups admitting a strictly finer non-discrete UFSS group topology. Unlike the minimal topologies, the locally minimal ones are always available on “large” groups. To support this line, we prove that a bounded abelian group G admits a non-discrete locally minimal and locally GTG group topology iff |G|?c.     

A Model-Theoretic Characterization of Countable Direct Sums of Finite Cyclic Groups #

We prove that an abelian group G is a countable direct sum of finite cyclic groups if and only if there exists a consistent existential theory Γ of abelian groups such that G is embeddable in every model of Γ.     

On purifiable subgroups in arbitrary abelian groups

We determine the structure of a p-pure[pure] hull of a p-purifiable [purifiable] subgroup of an arbitrary abelian group. Moreover, we prove that a subgroup A of an abelian group G is purifiable in G if and only if A is p-purifiable in G for every prime p. Using these results, we characterize the groups G for which all subgroups are purifiable in G. Furthermore, we establish several properties of purifiable subgroups.     

Fusions of Character Tables and Schur Rings of Abelian Groups

If the character table of a finite group H satisfies certain conditions, then the classes and characters of H can fuse to give the character table of a group G of the same order. We investigate the case where H is an abelian group. The theory is developed in terms of the S-rings of Schur and Wielandt. We discuss certain classes of p-groups which fuse from abelian groups and give examples of such groups which do not. We also show that a large class of simple groups do not fuse from abelian groups. The methods to show fusion include the use of extensions which are Camina pairs, but other techniques on S-rings are also developed.     

Fusions of Character Tables II: p-Groups

If the character table of a finite group H satisfies certain conditions, then the classes and characters of H can fuse to give the character table of a group G of the same order. We investigate the case where H is an abelian group. In a previous article, we gave examples of Camina pairs that fuse from abelian groups. In this article, we give more general examples of Camina triples that fuse from abelian groups. We use this result to give an example of a group which fuses from an abelian group, but which has a subgroup that does not. We also give an example of a powerful 2-group which does not fuse from an abelian group and of a regular 3-group which does not fuse from an abelian group.     

ON PRODUCTS OF GROUPS FOR WHICH NORMALITY IS A TRANSITIVE RELATION ON THEIR FRATTINI FACTOR GROUPS

Abstract

This paper is devoted to the study of groups with the property that the Frattini factor group is a T-group, i.e. a group in which every subnormal subgroup is normal. We give necessary and suffucient conditions for a direct product G = H x K of finite groups H and K to have such a property. Some structure theorems are also discussed.     

Purity and Self-Small Groups

Let A be an abelian group. A group B is A-solvable if the natural map Hom(A, B) ?  _E(A) A → B is an isomorphism. We study pure subgroups of A-solvable groups for a self-small group A of finite torsion-free rank. Particular attention is given to the case that A is in , the class of self-small mixed groups G with G/tG? ? ⁿ for some n < ω. We obtain a new characterization of the elements of , and demonstrate that  differs in various ways from the class  ? of torsion-free abelian groups of finite rank despite the fact that the quasi-category ?  is dual to a full subcategory of ?  ?.     

A Note on the Abelianization Functor

It is well known that the abelianization of a group G can be computed as the cokernel of the diagonal morphism (1_G, 1_G): G → G × G in the category of groups. We generalize this to arbitrary regular subtractive categories, among which are the category of groups, the category of topological groups, and the categories of other group-like structures. We also establish that an abelian category is the same as a regular subtractive category in which every monomorphism is a kernel of some morphism.     

Higher homotopy of groups definable in o-minimal structures

It is known that a definably compact group G is an extension of a compact Lie group L by a divisible torsion-free normal subgroup. We show that the o-minimal higher homotopy groups of G are isomorphic to the corresponding higher homotopy groups of L. As a consequence, we obtain that all abelian definably compact groups of a given dimension are definably homotopy equivalent, and that their universal covers are contractible.     

Essential subgroups of topological groups

We study the class Wof Hausdorff topological groups Gfor which the following two cardinal invariants coincide

ES(G)=min{|H|:H≤ Gdense and essential}

TD(G)=min{|H|:H≤ Gtotally dense}

We prove that W contains the following classes:locally compact abelian groups, compact connected groups, countably compact totally discon¬nected abelian groups, topologically simple groups, locally compact Abelian groups when endowed with their Bohr topology, totally minimal abelian groups and free Abelian topological groups. For all these classes we are also able to giv ean explicit computation of the common value of ESand TD.