Square of countably compact groups without non-trivial convergent sequences

We show in ZFC that the existence of a countably compact Abelian group without non-trivial convergent sequences implies the existence of a countably compact group whose square is not countably compact.This improves a result obtained by van Douwen in 1980: the existence of a countably compact Boolean group without non-trivial convergent sequences implies the existence of two countably compact groups whose product is not countably compact in ZFC.Hart and van Mill showed in 1991 the existence of a countably compact group whose square is not countably compact under Martin's Axiom for countable posets. We show that the existence of such an example does not depend on some form of Martin's Axiom.     

Divisibility of class numbers of imaginary quadratic fields whose discriminant has only three prime factors

We prove the existence of infinitely many imaginary quadratic fields whose discriminant has exactly three distinct prime factors and whose class group has an element of a fixed large order. The main tool we use is solving an additive problem via the circle method.     

Compact leaves of structurally stable foliations

We prove that any compact manifold whose fundamental group contains an abelian normal subgroup of positive rank can be represented as a leaf of a structurally stable suspension foliation on a compact manifold. In this case, the role of a transversal manifold can be played by an arbitrary compact manifold. We construct examples of structurally stable foliations that have a compact leaf with infinite solvable fundamental group which is not nilpotent. We also distinguish a class of structurally stable foliations each of whose leaves is compact and locally stable in the sense of Ehresmann and Reeb.     

Three examples of pseudocompact quasitopological groups

A quasitopological group is an abstract group with topology in which the inversion and all translations are continuous. We show that a pseudocompact quasitopological group of countable cellularity need not be a Moscow space. Then we present an example of two pseudocompact quasitopological groups whose product fails to be pseudocompact, and of a pseudocompact quasitopological group that contains an infinite discrete subgroup.     

真商群为s-自对偶群的有限p-群

如果有限群G的每个子群与G的某个商群同构,则称群G为s-自对偶群.如果s-自对偶群G的每个商群与G的某个子群同构,则称群G为自对偶群.本文分类了每个真商群均为s-自对偶群的有限p-群.作为推论,本文还分类了每个真截段均为s-自对偶群的有限p-群,每个真商群均为自对偶群的有限p-群,以及每个真截段均为自对偶群的有限p-群...     

具有一个T．I．Sylow 2-子群的有限群的类保持Coleman自同构

设G是一个有限群,它的Sylow 2-子群是T.I.集,证明了如果G的2的方幂阶类保持自同构在G任意的Sylow子群上的限制等于G的某个内自同构的限制,则它一定是一个内自同构.对这样的自同构的研究是由整群环的同构问题所引起的.     

Algebraic monoids with affine unit group are affine

In this short paper we prove that any irreducible algebraic monoid whose unit group is an affine algebraic group is affine.     

FC groups whose periodic parts can be embedded in direct products of finite groups

In this note are considered FC groups whose periodic parts can be embedded in direct products of finite groups. It is shown that if the periodic part of an FC group G can be embedded in the direct product of its finite factor groups with respect to the normal subgroups of G whose intersection is the trivial subgroup, then G/Z (G) is a subgroup of a direct product of finite groups. It is also shown that if the periodic part of an FC group G is a group without a center, then G can be embedded in a direct product of finite groups without centers and a torsion-free Abelian group. Translated from Matematicheskie Zametki, Vol. 21, No. 1, pp. 9–20, January, 1977. The author is thankful to the referee for making many valuable remarks.     

Normality of the Twistor Space of a 5-Manifold with an Irreducible SO(3)-Structure

Mediterranean Journal of Mathematics - A manifold with an irreducible SO(3)-structure is a 5-manifold M whose structure group can be reduced to the group SO(3), non-standardly imbedded in SO(5)....     

Spectra of elements in the range of strongly continuous unitary representations of Lie groups

The main result proved in this work is that, for a strongly continuous unitary representation of a Lie group, we have the following dichotomy: The representation is norm continuous or on an open neighborhood of the unit (which is the whole group for a weakly exponential Lie group), there is a residual set of elements whose range admits the entire torus as spectrum.     

一类可解的（q）－群

有限群Ｇ叫（ｑ）－群，如果Ｇ中每个次正规子群均为拟正规子群，群Ｇ叫Ｅｑ－群，若Ｇ中每个子群在Ｇ中拟正规或自正规，有限群Ｇ叫内Ｅｑ－群，如果Ｇ本身不是Ｅｑ－群，但Ｇ的每个真子群是Ｅｑ－群，本文确定了Ｅｑ－群的结构与内Ｅｑ－群的分类．     

An Infinite Presentation of the Torelli Group

In this paper, we construct an infinite presentation of the Torelli subgroup of the mapping class group of a surface whose generators consist of the set of all “separating twists”, all “bounding pair maps”, and all “commutators of simply intersecting pairs” and whose relations all come from a short list of topological configurations of these generators on the surface. Aside from a few obvious ones, all of these relations come from a set of embeddings of groups derived from surface groups into the Torelli group. In the process of analyzing these embeddings, we derive a novel presentation for the fundamental group of a closed surface whose generating set is the set of all simple closed curves.     

On periodic groups with prescribed orders of elements

This is a survey of results and open problems concerning the structure of periodic group with prescribed orders of their elements. In particular, an outline of a proof of the following recent result is given: every periodic group whose set of element orders coincides with {1,2,3,4,8} is locally finite.     

Real Algebraically Maximal Varieties

For real algebraic varieties whose real algebraic cohomology group is maximal, a canonical homomorphism is constructed from the cohomology group of the set of complex points into the cohomology group of the set of real points, and then it is proved that this homomorphism is an isomorphism.     

Class-Preserving Coleman Automorphisms of Finite Groups

 Let G be a finite group whose Sylow 2-subgroups are either cyclic, dihedral, or generalized quaternion. It is shown that a class-preserving automorphism of G of order a power of 2 whose restriction to any Sylow subgroup of G equals the restriction of some inner automorphism of G is necessarily an inner automorphism. Interest in such automorphisms arose from the study of the isomorphism problem for integral group rings, see [6, 7, 13, 14].     

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.     

Connections defined in bundles by the action of infinite Lie groups

Given a principal bundle P=X×V with an arbitrary connection, whose base space X is diffeomorphic to the group manifold of some Lie group, the author constructs an infinite Lie group ^H whose action defines the geometrical structure of P. The structure equation of P is a necessary condition for the existence of a group ^H defining the given connection in P.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 43, No. 12, pp. 1599–1603, December, 1991.     

Power graphs of (non)orientable genus two

The power graph of a finite group is the graph whose vertex set is the group, two distinct elements being adjacent if one is a power of the other. In this article, we classify the finite groups whose power graphs have (non)orientable genus two.     

Orthonormal bases of symmetry classes with computer-generated examples

A method is given for constructing an orthonormal basis of a symmetry class of tensor whose associated group is the full symmetric group. This construction has been used to write a computer program which efficiently calculates a basis for any such symmetry class. Two examples which illustrate both the method and the program are included.     

On the genera of graphs of group presentations. II

The genus of the graph of any less-than-one-24th group, in the sense of Greendlinger, all of whose relators are of equal lengths, is shown to be decidable via the existence of an algorithm.