## Abstract

Let *G*, *H* be groups and let $\kappa $ be a cardinal. A bijection $f:G\to H$ is called an 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 γ be cardinals such that ${\mathrm{\aleph}}_{0}<\kappa \le \gamma $.
We prove that any two Abelian groups of cardinality γ are $\kappa $-asymorphic, but the free group of rank γ 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 [9] that if $\gamma =\kappa $ and $\kappa $ is regular, then any two groups of cardinality $\kappa $ are $\kappa $-asymorphic.

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