On asymorphisms of groups

TL;DR

This paper introduces the concept of asymorphisms between groups, explores their properties for Abelian and free groups, and establishes conditions under which groups are or are not asymorphic based on their cardinalities.

Contribution

It defines the notion of asymorphism for groups and proves new results about when groups of certain cardinalities are asymorphic or not.

Findings

Any two Abelian groups of the same uncountable cardinality are $oldsymbol{ ext{kappa}}$-asymorphic.

A free group of uncountable rank is not $oldsymbol{ ext{kappa}}$-asymorphic to an Abelian group under certain conditions.

The paper extends known results about asymorphisms for groups of regular and singular cardinals.

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.