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# Journal of Group Theory

Editor-in-Chief: Parker, Christopher W. / Wilson, John S.

Managing Editor: Khukhro, Evgenii I. / Kramer, Linus

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# On asymorphisms of groups

Igor Protasov
• Department of Cybernetics, Kyiv University, Volodymyrska 64, 01033, Kyiv, Ukraine
• Email:
/ Serhii Slobodianiuk
• Department of Mechanics and Mathematics, Kyiv University, Volodymyrska 64, 01033, Kyiv, Ukraine
• Email:
Published Online: 2016-09-14 | DOI: https://doi.org/10.1515/jgth-2016-0038

## 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 $X\in {\left[G\right]}^{<\kappa }$, $Y\in {\left[H\right]}^{<\kappa }$, there exist ${X}^{\prime }\in {\left[G\right]}^{<\kappa }$, ${Y}^{\prime }\in {\left[H\right]}^{<\kappa }$ such that for all $x\in G$ and $y\in H$, we have $f\left(Xx\right)\subseteq {Y}^{\prime }f\left(x\right)$, ${f}^{-1}\left(Yy\right)\subseteq {X}^{\prime }{f}^{-1}\left(y\right)$. For a set S, ${\left[S\right]}^{<\kappa }$ denotes the set $\left\{{S}^{\prime }\subseteq S:|{S}^{\prime }|<\kappa \right\}$. 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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Revised: 2016-08-11

Published Online: 2016-09-14

Published in Print: 2017-03-01

Citation Information: Journal of Group Theory, ISSN (Online) 1435-4446, ISSN (Print) 1433-5883, Export Citation