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The superextension $lambda(X)$ of a set $X$ consists of all maximal linked families on $X$. Any associative binary operation $*: Xtimes X to X$ can be extended to an associative binary operation $*: lambda(X)timeslambda(X)tolambda(X)$. In the paper we study isomorphisms of superextensions of groups and prove that two groups are isomorphic if and only if their superextensions are isomorphic. Also we describe the automorphism groups of superextensions of all groups of cardinality $leq 5$.
A family $mathcal L$ of subsets of a set $X$ is called linked if $Acap B eemptyset$ for any $A,Binmathcal L$. A linked family $mathcal M$ of subsets of $X$ is maximal linked if $mathcal M$ coincides with each linked family $mathcal L$ on $X$ that con
Given a group $X$ we study the algebraic structure of the compact right-topological semigroup $lambda(X)$ consisting of maximal linked systems on $X$. This semigroup contains the semigroup $beta(X)$ of ultrafilters as a closed subsemigroup. We constr
We show that Out(G) is residually finite if G is a one-ended group that is hyperbolic relative to virtually polycyclic subgroups. More generally, if G is one-ended and hyperbolic relative to proper residually finite subgroups, the group of outer auto
In this article we present an extensive survey on the developments in the theory of non-abelian finite groups with abelian automorphism groups, and pose some problems and further research directions.