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Productivity of the Zariski topology on groups

Dikran N. Dikranjan, D. Toller (2013)

Commentationes Mathematicae Universitatis Carolinae

This paper investigates the productivity of the Zariski topology G of a group G . If 𝒢 = { G i i I } is a family of groups, and G = i I G i is their direct product, we prove that G i I 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., G = G 1 × G 2 . If e 2 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 × { e 2 } is an elementary algebraic subset of G 1 × G 2 for every group G 1 . We show among others, that Δ is stable...

Pronormal and subnormal subgroups and permutability

James Beidleman, Hermann Heineken (2003)

Bollettino dell'Unione Matematica Italiana

We describe the finite groups satisfying one of the following conditions: all maximal subgroups permute with all subnormal subgroups, (2) all maximal subgroups and all Sylow p -subgroups for p < 7 permute with all subnormal subgroups.

Pure virtual braids homotopic to the identity braid

H. A. Dye (2009)

Fundamenta Mathematicae

Two virtual link diagrams are homotopic if one may be transformed into the other by a sequence of virtual Reidemeister moves, classical Reidemeister moves, and self crossing changes. We recall the pure virtual braid group. We then describe the set of pure virtual braids that are homotopic to the identity braid.

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