On asymorphisms of groups
Igor Protasov, Serhii Slobodianiuk

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 , be groups and be a cardinal. A bijection is caled on asymorphism if, for any , , there exist , such that for all and , we have , . For a set , denotes the set . Let and be cardinals such that . We prove that any two Abelian groups of cardinality are -asymorphic, but the free group of rank is not -asymorphic to an Abelian group provided that either or and is a singular cardinal. It is known [7] that if and is regular then any two groups of cardinality are -asymorphic.
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