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It is proved that, in certain subgroups of direct products of countable groups, the property of being an unconditionally closed set coincides with that of being an algebraic set. In particular, these properties coincide in all Abelian groups.
Families of unconditionally $tau$-closed and $tau$-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 i
We investigate the class $mathcal{MN}$ of groups with the property that all maximal subgroups are normal. The class $mathcal{MN}$ appeared in the framework of the study of potential counter-examples to the Andrews-Curtis conjecture. In this note we g
The article deals with profinite groups in which the centralizers are abelian (CA-groups), that is, with profinite commutativity-transitive groups. It is shown that such groups are virtually pronilpotent. More precisely, let G be a profinite CA-group
The article deals with profinite groups in which centralizers are virtually procyclic. Suppose that G is a profinite group such that the centralizer of every nontrivial element is virtually torsion-free while the centralizer of every element of infin
For an element $g$ of a group $G$, an Engel sink is a subset $mathcal{E}(g)$ such that for every $ xin G $ all sufficiently long commutators $ [x,g,g,ldots,g] $ belong to $mathcal{E}(g)$. We conjecture that if $G$ is a profinite group in which every