Stokes Cocycle and Differential Galois Groups

The classification of germs of ordinary linear differential systems with meromorphic coefficients at 0 under convergent gauge transformations and fixed normal form is essentially given by the non-Abelian 1-cohomology set of Malgrange–Sibuya. (Germs themselves are actually classified by a quotient of this set.) It is known that there exists a natural isomorphism h between a unipotent Lie group (called the Stokes group) and the 1-cohomology set of Malgrange–Sibuya; the inverse map which consists of choosing, in each cohomology class, a special cocycle called a Stokes cocycle is proved to be natural and constructive. We survey here the definition of the Stokes cocycle and give a combinatorial proof for the bijectivity of h. We state some consequences of this result, such as Ramis, density theorem in linear differential Galois theory; we note that such a proof based on the Stokes cocycle theorem and the Tannakian theory does not require any theory of (multi-)summation.     

Butterflies II: Torsors for 2-group stacks

We study torsors over 2-groups and their morphisms. In particular, we study the first non-abelian cohomology group with values in a 2-group. Butterfly diagrams encode morphisms of 2-groups and we employ them to examine the functorial behavior of non-abelian cohomology under change of coefficients. We re-interpret the first non-abelian cohomology with coefficients in a 2-group in terms of gerbes bound by a crossed module. Our main result is to provide a geometric version of the change of coefficients map by lifting a gerbe along the “fraction” (weak morphism) determined by a butterfly. As a practical byproduct, we show how butterflies can be used to obtain explicit maps at the cocycle level. In addition, we discuss various commutativity conditions on cohomology induced by various degrees of commutativity on the coefficient 2-groups, as well as specific features pertaining to group extensions.     

Transitive permutation groups in which all derangements are involutions

Let G be a transitive permutation group in which all derangements are involutions. We prove that G is either an elementary abelian 2-group or is a Frobenius group having an elementary abelian 2-group as kernel. We also consider the analogous problem for abstract groups, and we classify groups G with a proper subgroup H such that every element of G not conjugate to an element of H is an involution.     

On groups whose element orders divide 6 and 7

We prove that a group whose element orders divide 6 and 7 either is locally finite or an extension of a nontrivial elementary abelian 2-group by a group without involutions.     

The <Emphasis Type="Italic">C</Emphasis>-topology on lattice-ordered groups

Let A be a lattice-ordered group. Gusi′c showed that A can be equipped with a C-topology which makes A into a topological group. We give a generalization of Gusi′c's theorem,and reveal the very nature of a "C-group" of Gusi′c in this paper. Moreover,we show that the C-topological groups are topological lattice-ordered groups,and prove that every archimedean lattice-ordered vector space is a T2 topological lattice-ordered vector space under the C-topology. An easy example shows that a C-group need not be T2....     

Higher dimensional isoperimetric functions in hyperbolic groups

We introduce the notion of an -combing and use it to show that hyperbolic groups satisfy linear isoperimetric inequalities for filling real cycles in each positive dimension. S. Gersten suggested the concept of metabolicity (over or ) for groups which implies hyperbolicity. Metabolicity admits several equivalent definitions: by vanishing of -cohomology, using combings, and others. We prove several criteria for a group to be hyperbolic, -metabolicity being among them. In particular, a finitely presented group G is hyperbolic iff for any normed vector space V and any . Received December 9, 1998     

Reflexivity of prodiscrete topological groups

We study the duality properties of two rather different classes of subgroups of direct products of discrete groups (protodiscrete groups): P-groups, i.e., topological groups such that countable intersections of its open subsets are open, and protodiscrete groups of countable pseudocharacter (topological groups in which the identity is the intersection of countably many open sets). It was recently shown by the same authors that the direct product Π of an arbitrary family of discrete Abelian groups becomes reflexive when endowed with the ω-box topology. This was the first example of a non-discrete reflexive P-group. Here we present a considerable generalization of this theorem and show that every product of feathered (equivalently, almost metrizable) Abelian groups equipped with the P-modified topology is reflexive. In particular, every locally compact Abelian group with the P-modified topology is reflexive. We also examine the reflexivity of dense subgroups of products Π with the P-modified topology and obtain the first examples of non-complete reflexive P-groups. We find as well that the better behaved class of prodiscrete groups (complete protodiscrete groups) of countable pseudocharacter contains non-reflexive members—any uncountable bounded torsion Abelian group G of cardinality ω² supports a topology τ such that (G,τ) is a non-reflexive prodiscrete group of countable pseudocharacter.     

On positive and constructive groups

Considering a group with unique roots (i.e., an R-group), we give a sufficient condition for the existence of a positive (constructive) enumeration with respect to which the isolator of the commutant is computable. Basing on it, we prove the constructivizability of an R-group that admitting a positive enumeration for which the dimension of the commutant is finite. We obtain a necessary and sufficient condition of constructivizability for a torsion-free nilpotent group for which the dimension of the commutant is finite.     

On the periodic groups saturated by direct products of finite simple groups

We prove the local finiteness of a periodic group G saturated by direct products of an elementary abelian 2-group of fixed order and the simple groups L ₂(q) under condition that G contains an element of order 4.     

Irreducible and transitive locally-nilpotent by abelian finitary groups

We show that a locally-nilpotent by abelian group that is either a transitive finitary permutation group on an infinite set or an irreducible finitary skew linear group of infinite dimension, is a p-group for some suitably chosen prime p.     

Dieudonné Completion and PT-Groups

We consider the classes of PT-groups, strong PT-groups, completion friendly groups, and Moscow groups introduced by Arhangel’skii for the study of the Dieudonné completion of topological groups. We show that every subgroup H of a Lindel?f P-group is a PT-group, and that H is a strong PT-group iff it is \mathbb R{\mathbb R}-factorizable. Assuming CH, we prove that every ω-narrow P-group is a PT-group. Several results regarding products of PT-groups and \mathbb R{\mathbb R}-factorizable groups are established as well. We prove that the product of a Lindel?f group and an arbitrary subgroup of a Lindel?f Σ-group is completion friendly, and the same conclusion is valid for the product of an \mathbb R{\mathbb R}-factorizable P-group with an almost metrizable group.     

Finite groups in which normality is a transitive relation

Let G be a finite group. We say that G is a T₀-group, if its Frattini quotient group G/F(G)G/\Phi (G) is a T-group, where by a T-group we mean a group in which every subnormal subgroup is normal. We determine the structure of a non T₀-group G all of whose proper subgroups are T₀-groups.     

Virtually free pro-p groups

We prove that in the category of pro-p groups any finitely generated group G with a free open subgroup splits either as an amalgamated free product or as an HNN-extension over a finite p-group. From this result we deduce that such a pro-p group is the pro-p completion of a fundamental group of a finite graph of finite p-groups.     

On cocycle twisting of compact quantum groups

We provide a class of examples of compact quantum groups and unitary 2-cocycles on them, such that the twisted quantum groups are non-compact, but still locally compact quantum groups (in the sense of Kustermans and Vaes). This also gives examples of cocycle twists where the underlying C^∗-algebra of the quantum group changes.     

Almost Butler groups

Generalizing the notion of the almost free group we introduce almost Butler groups. An almost B ₂-group G of singular cardinality is a B ₂-group. Since almost B ₂-groups have preseparative chains, the same result in regular cardinality holds under the additional hypothesis that G is a B ₁-group. Some other results characterizing B ₂-groups within the classes of almost B ₁-groups and almost B ₂-groups are obtained. A theorem of [BR] stating that a group G of weakly compact cardinality having a -filtration consisting of pure B ₂-subgroup is a B ₂-group appears as a corollary.     

Hyperbolic groups have flat-rank at most 1

The flat-rank of a totally disconnected, locally compact group G is an integer, which is an invariant of G as a topological group. We generalize the concept of hyperbolic groups to the topological context and show that a totally disconnected, locally compact, hyperbolic group has flat-rank at most 1. It follows that the simple totally disconnected locally compact groups constructed by Paulin and Haglund have flat-rank at most 1.     

Topological nearrings whose additive groups are Euclidean

Let (G,+) be a group with a locally compact Hausdorff topology for which the binary operation + is continuous. Those, binary operation * onG for which (G, +, *) is a topological nearring are described. In the case whereG is abelian, those binary operations * for which (G, +, *) is a topological ring are also described. Versions of these results are then obtained in the special case where the group is the topological Euclideann-group,R ⁿ. A family of binary operations * for which (R ⁿ, +, *)_is a topological nearring is then investigated in some detail. Most of these nearrings turn out to be planar. Their ideals are completely determined and we characterize those nearrings which are simple. The multiplicative semi-groups (R ⁿ, *) of these nearrings are then investigated. Green's relations are completely determined and it is shown that a number of familiar properties of semigroups are equivalent for these particular semigroups. Finally, all those binary operations * for which (R, +, *) is a topological nearring are completely described. It is determined when any two of these nearrings are isomorphic and for each of these nearrings, its automorphism group, is completely determined.     

Minimal Number of Generators and Minimum Order of a Non-Abelian Group Whose Elements Commute with Their Endomorphic Images

A group in which every element commutes with its endomorphic images is called an “E-group″. If p is a prime number, a p-group G which is an E-group is called a “pE-group″. Every abelian group is obviously an E-group. We prove that every 2-generator E-group is abelian and that all 3-generator E-groups are nilpotent of class at most 2. It is also proved that every infinite 3-generator E-group is abelian. We conjecture that every finite 3-generator E-group should be abelian. Moreover, we show that the minimum order of a non-abelian pE-group is p ⁸ for any odd prime number p and this order is 2⁷ for p = 2. Some of these results are proved for a class wider than the class of E-groups.