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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...

Products of Conjugacy Classes in a Free Group: a Counterexample.

S.M. Gersten (1986)

Mathematische Zeitschrift

Products of conjugacy classes in finite unitary groups $G U (3, q^{2})$ and $S U (3, q^{2})$

S.Yu. Orevkov (2013)

Annales de la faculté des sciences de Toulouse Mathématiques

For the groups $G U (3)$ , $S U (3)$ , $G L (3)$ , $S U (3)$ over a finite field we solve the class product problem, i.e., we give a complete list of $m$ -tuples of conjugacy classes whose product does not contain the identity matrix.

Products of conjugacy classes in perfect linear groups. Extended covering number.

Gordeev, N.L. (2005)

Zapiski Nauchnykh Seminarov POMI

Products of trees, lattices and simple groups.

Mozes, Shahar (1998)

Documenta Mathematica

Profinite completions of the fundamental group of the Klein bottle

Hugo H. Torriani (1985)

Czechoslovak Mathematical Journal

Projectable kernel of a lattice ordered group

Ján Jakubik (1982)

Banach Center Publications

Projective simplicity of groups of rational points of simply connected algebraic groups defined over number fields

G. Tomano (1990)

Banach Center Publications

Projectivities of free products

Charles S. Holmes (1969)

Rendiconti del Seminario Matematico della Università di Padova

Projektivitätstypen torsionsfreier abelscher Gruppen vom Rang 1

Uwe Ostendorf (1991)

Rendiconti del Seminario Matematico della Università di Padova

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.

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