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This paper investigates the productivity of the Zariski topology $ℨ_{G}$ of a group $G$ . If $𝒢 = {G_{i} ∣ i \in I}$ is a family of groups, and $G = \prod_{i \in I} G_{i}$ is their direct product, we prove that $ℨ_{G} \subseteq \prod_{i \in I} ℨ_{G_{i}}$ . This inclusion can be proper in general, and we describe the doubletons $𝒢 = {G_{1}, G_{2}}$ of abelian groups, for which the converse inclusion holds as well, i.e., $ℨ_{G} = ℨ_{G_{1}} \times ℨ_{G_{2}}$ . If $e_{2} \in G_{2}$ is the identity element of a group $G_{2}$ , we also describe the class $Δ$ of groups $G_{2}$ such that $G_{1} \times {e_{2}}$ is an elementary algebraic subset of $G_{1} \times G_{2}$ for every group $G_{1}$ . We show among others, that $Δ$ is stable...

Pure braid subgroups of braided Thompson's groups.

T. Brady, J. Burillo, S. Cleary, M. Stein (2008)

Publicacions Matemàtiques

Q-perfect groups and universal Q-central extensions.

Ronald Brown (1990)

Publicacions Matemàtiques

Using results of Ellis-Rodríguez Fernández, an explicit description by generators and relations is given of the mod q Schur multiplier, and this is shown to be the kernel of a universal q-central extension in the case of a q-perfect group, i.e. one which is generated by commutators and q-th powers. These results generalise earlier work [by] K. Dennis and Brown-Loday.

Quasigroups arisen by right nuclear extension

Péter T. Nagy, Izabella Stuhl (2012)

Commentationes Mathematicae Universitatis Carolinae

The aim of this paper is to prove that a quasigroup $Q$ with right unit is isomorphic to an $f$ -extension of a right nuclear normal subgroup $G$ by the factor quasigroup $Q / G$ if and only if there exists a normalized left transversal $Σ \subset Q$ to $G$ in $Q$ such that the right translations by elements of $Σ$ commute with all right translations by elements of the subgroup $G$ . Moreover, a loop $Q$ is isomorphic to an $f$ -extension of a right nuclear normal subgroup $G$ by a loop if and only if $G$ is middle-nuclear, and there exists...

Quotients of subdirect products of groups.

Gorchakov, Yu.M. (2009)

Sibirskij Matematicheskij Zhurnal

Relative property (T) and linear groups

Talia Fernós (2006)

Annales de l’institut Fourier

Relative property (T) has recently been used to show the existence of a variety of new rigidity phenomena, for example in von Neumann algebras and the study of orbit-equivalence relations. However, until recently there were few examples of group pairs with relative property (T) available through the literature. This motivated the following result: A finitely generated group $Γ$ admits a special linear representation with non-amenable $R$ -Zariski closure if and only if it acts on an Abelian group $A$ (of...

Representations of Certain Verbal Wreath Products by Matrices.

Michael W. Thomson (1979)

Mathematische Zeitschrift

Representations of the generalized symmetric groups.

Can, Himmet (1996)

Beiträge zur Algebra und Geometrie

Residually finite PSP-groups

A. Mohammadi Hassanabadi (1999)

Rendiconti del Seminario Matematico della Università di Padova

Restricted subgroups of wreath products of groups

C. H. Houghton (1976)

Compositio Mathematica

Schreier loops

Péter T. Nagy, Karl Strambach (2008)

Czechoslovak Mathematical Journal

We study systematically the natural generalization of Schreier's extension theory to obtain proper loops and show that this construction gives a rich family of examples of loops in all traditional common, important loop classes.

Schreier type theorems for bicrossed products

Ana Agore, Gigel Militaru (2012)

Open Mathematics

We prove that the bicrossed product of two groups is a quotient of the pushout of two semidirect products. A matched pair of groups (H;G; α; β) is deformed using a combinatorial datum (σ; v; r) consisting of an automorphism σ of H, a permutation v of the set G and a transition map r: G → H in order to obtain a new matched pair (H; (G; *); α′, β′) such that there exists a σ-invariant isomorphism of groups H α⋈β G ≅H α′⋈β′ (G, *). Moreover, if we fix the group H and the automorphism σ ∈ Aut H then...

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Displaying 101 – 120 of 207

On Virtual Properties and Group Extensions.

Order Reduced Reidemeister-Schreier Subgroup Presentations and Applications.

Plane curves and their fundamental groups: Generalizations of Uludağ's construction.

Primary graded groups with systems of complemented subgroups.

Prodotti di sottogruppi mutuamente permutabili

Productivity of the Zariski topology on groups