On asymorphisms of groups

Abstract

Let $G$, $H$ be groups and $\kappa$ be a cardinal. A bijection $f:G\to H$ is caled on asymorphism if, for any $X\in[G]^{<\kappa}$, $Y\in[H]^{<\kappa}$, there exist $X'\in[G]^{<\kappa}$, $Y'\in[H]^{<\kappa}$ such that for all $x\in G$ and $y\in H$, we have $f(Xx)\subseteq Y'f(x)$, $f^{-1}(Yy)\subseteq X'f^{-1}(y)$. For a set $S$, $[S]^{<\kappa}$ denotes the set $\{S'\subseteq S: |S'|<\kappa\}$. Let $\kappa$ and $\gamma$ be cardinals such that $\aleph_0<\kappa\le\gamma$. We prove that any two Abelian groups of cardinality $\gamma$ are $\kappa$-asymorphic, but the free group of rank $\gamma$ is not $\kappa$-asymorphic to an Abelian group provided that either $\kappa<\gamma$ or $\kappa=\gamma$ and $\kappa$ is a singular cardinal. It is known [7] that if $\gamma = \kappa$ and $\kappa$ is regular then any two groups of cardinality $\kappa$ are $\kappa$-asymorphic.