On left conjugacy closed loops with a nucleus of index two

A loop Q is said to be left conjugacy closed (LCC) if the left translations form a set of permutations that is closed under conjugation. Loops in which the left and middle nuclei coincide and are of index 2 are necesarilly LCC, and they are constructed in the paper explicitly. LCC loops Q with the right nucleus G of index 2 offer a larger diversity, but that is associated with the level of commutativity of G (amongst others, the centre of G has to be nontrivial). On the other hand, for each m ≥ 2 one can construct an LCC loop Q of order 2m in such a way that its left nucleus is trivial, and the right nucleus if of order m. If Q is involutorial, then it is a Bol loop. Work supported by institutional grant MSM 113200007 and by Grant Agency of Czech Republic, Grant 201/02/0594. The paper was written while the author was visiting Universitaet Hamburg in January 2004.     

Topological loops with three-dimensional solvable left translation group

We classify all connected topological loops having a three-dimensional solvable Lie group G as the group topologically generated by their left translations. It is surprising that to the non-nilpotent Lie group G having precisely one one-dimensional normal subgroup there are topological but no differentiable strongly left alternative loops.     

Character tables and normal left coideal subalgebras

We continue studying properties of semisimple Hopf algebras H over algebraically closed fields of characteristic 0 resulting from their generalized character tables. We show that the generalized character table of H reflects normal left coideal subalgebras of H. These are the Hopf analogues of normal subgroups in the sense that they arise from Hopf quotients. We apply these ideas to prove Hopf analogues of known results in group theory. Among the rest we prove that columns of the character table are orthogonal and that all entries are algebraic integers. We analyze ‘semi-kernels’ and their relations to the character table. We prove a full analogue of the Burnside–Brauer theorem for almost cocommutative H. We also prove the Hopf algebras analogue of the following (Burnside) theorem: If G is a non-abelian simple group then {1} is the only conjugacy class of G which has prime power order.     

Affine Reductive Spaces of Small Dimension and Left A-Loops

In this paper we determine the at least 4-dimensional affine reductive homogeneous manifolds for an at most 9-dimensional simple Lie group or an at most 6-dimensional semi-simple Lie group. Those reductive spaces among them which admit a sharply transitive differentiable section yield local almost differentiable left A-loops. Using this we classify all global almost differentiable left A-loops L having either a 6-dimensional semi-simple Lie group or the group as the group topologically generated by their left translations. Moreover, we determine all at most 5-dimensional left A-loops L with as the group topologically generated by their left translations.     

Bol loops as sections in semi-simple Lie groups of small dimension

Using the relations between the theory of differentiable Bol loops and the theory of affine symmetric spaces we classify all connected differentiable Bol loops having an at most nine-dimensional semi-simple Lie group as the group topologically generated by their left translations. We show that all these Bol loops are isotopic to direct products of Bruck loops of hyperbolic type or to Scheerer extensions of Lie groups by Bruck loops of hyperbolic type.This paper was supported by DAAD.     

Coverings of topological loops

In this paper, we show that in some cases, no proper covering of a locally compact group topologically generated by left translations of a topological loop can occur as the group topologically generated by left translations of a topological loop. __________ Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 22, Algebra and Geometry, 2004.     

Explicit left orders on free groups extending the lexicographic order on free monoids

For every finitely generated free group, we construct an explicit left order extending the lexicographic order on the free monoid generated by the positive letters. The order is defined by a left, free action on the orbit of 0 of a free group of piecewise linear homeomorphisms of the line. The membership in the positive cone is decidable in linear time in the length of the input word. The positive cone forms a context-free language closed under word reversal.     

Translation groups of Steiner loops

If the order of any product of two different translations of a finite Steiner quasigroup of size n>3 is odd, then the group G generated by the translations of the corresponding Steiner loop of order n+1 contains the alternating group of degree n+1.     

Characterizing Translations on Groups by Cosets of Their Subgroups

On a group, constant functions and left translations by group elements map left cosets into left cosets for every subgroup. We determine classes of groups for which this property of preserving cosets characterizes constants and translations, e.g., finite non-abelian groups that are perfect, partitioned, primitive, or generated by elements of prime order p. For certain classes of groups we construct other coset-preserving functions, in particular, power endomorphisms and functions defined in terms of the subgroup lattice.     

Continuity in topological groups

We prove that, when G is a group equipped with a Baire and metrizable topology, if there is a second category dense subset S of G such that the right translations ρs and ρs⁻¹ are continuous for all s∈S and each left translation λs, s∈G, is almost-continuous (defined below) on a residual subset of G, then G is a topological group. Among other consequences, this yields that when G is a second countable locally compact right topological group, its topological centre is a topological group.     

Unconditionally τ-closed and τ-algebraic sets in groups

Families of unconditionally τ-closed and τ-algebraic sets in a group are defined, which are natural generalizations of unconditionally closed and algebraic sets defined by Markov. A sufficient condition for the coincidence of these families is found. In particular, it is proved that these families coincide in any group of cardinality at most τ. This result generalizes both Markov's theorem on the coincidence of unconditionally closed and algebraic sets in a countable group (as is known, they may be different in an uncountable group) and Podewski's theorem on the topologizability of any ungebunden group.     

The holomorphic automorphism group of the complex disk

The group of all holomorphic automorphisms of the complex unit disk consists of Möbius transformations involving translation-like holomorphic automorphisms and rotations. The former are calledgyrotranslations. As opposed to translations of the complex Plane, which are associative-commutative operations forming a group, gyrotranslations of the complex unit disk fail to form a group. Rather, left gyrotranslations are gyroassociative-gyrocommutative operations forming agyrogroup.     

Note on a Result of Kerman and Weit II

Let G be a locally compact topological group, equipped with a fixed left Haar measure μ. We show that if f is a compactly supported real valued continuous function on G which has a unique maximum or a unique minimum at a point in G, then the space generated by the span of left translations of {f ⁿ ∣n=1,2,3,…} is dense in L ^p (G,μ), 1≤p<∞, in the space of continuous functions, continuous compactly supported functions and in the space of continuous functions vanishing at ∞. Similar results are true when the group G is substituted by G-spaces with compact isotropy group.     

Generators and relations for subsemigroups via boundaries in Cayley graphs

Given a finitely generated semigroup S and subsemigroup T of S, we define the notion of the boundary of T in S which, intuitively, describes the position of T inside the left and right Cayley graphs of S. We prove that if S is finitely generated and T has a finite boundary in S then T is finitely generated. We also prove that if S is finitely presented and T has a finite boundary in S then T is finitely presented. Several corollaries and examples are given.     

A note on the left and right translations in L-topological groups

In this note, we show by a counterexample that a proposition on L-homeomorphism property of the left and right translations in L-topological groups given by Bayoumi [On initial and final L-topological groups, Fuzzy Sets and Systems 156 (2005) 43–54] is incorrect, and offer its modified version.     

On the continuity of left translations in the LUC-compactification

Consider the continuity of left translations in the LUC-compactification GLUC of a locally compact group G. For every X⊆G, let κ(X) be the minimal cardinality of a compact covering of X in G. Let U(G) be the points in GLUC that are not in the closure of any X⊆G with κ(X)<κ(G). We show that the points at which no left translation in U(G) is continuous are dense in U(G). This result is a generalization of a theorem by van Douwen concerning discrete groups. We obtain a new proof for the fact that the topological center of GLUC?G is empty.     

Bounded Engel elements in groups satisfying an identity

We prove that a residually finite group G satisfying an identity \(w\equiv 1\) and generated by a commutator closed set X of bounded left Engel elements is locally nilpotent. We also extend such a result to locally graded groups, provided that X is a normal set. As an immediate consequence, we obtain that a locally graded group satisfying an identity, all of whose elements are bounded left Engel, is locally nilpotent.     

An asymmetric Ellis theorem

In 1957 Robert Ellis proved that a group with a locally compact Hausdorff topology T making all translations continuous also has jointly continuous multiplication and continuous inversion, and is thus a topological group. The theorem does not apply to locally compact asymmetric spaces such as the reals with addition and the topology of upper open rays. We first show a bitopological Ellis theorem, and then introduce a generalization of locally compact Hausdorff, called locally skew compact, and a topological dual, Tk, to obtain the following asymmetric Ellis theorem which applies to the example above:Whenever (X,⋅,T) is a group with a locally skew compact topology making all translations continuous, then multiplication is jointly continuous in both (X,⋅,T) and (X,⋅,Tk), and inversion is a homeomorphism between (X,T) and (X,Tk).This generalizes the classical Ellis theorem, because T=Tk when (X,T) is locally compact Hausdorff.     

Classifying subcategories of finitely generated modules over a Noetherian ring

For R a commutative Noetherian ring, wide and Serre subcategories of finitely generated R-modules have been classified by their support. This paper studies general torsion classes and introduces narrow subcategories. These are closed under fewer operations than wide and Serre subcategories, but still for finitely generated R-modules both narrow subcategories and torsion classes are classified using the same support data. Although for finitely generated R-modules all four kinds of subcategories coincide, they do not coincide in the larger category of all R-modules.