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 $X^{\prime }\in [G{]}^{<\kappa }$, $Y^{\prime }\in [H{]}^{<\kappa }$ such that for all $x\in G$ and $y\in H$, we have $f(Xx)\subseteq Y^{\prime }f(x)$, $f^{-1}(Yy)\subseteq X^{\prime }{f}^{-1}(y)$. For a set $S$, $[S{]}^{<\kappa }$ denotes the set $\{S^{\prime }\subseteq S:|S^{\prime }|<\kappa \}$. 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.