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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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Online
ISSN
1435-4446
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Volume 20, Issue 2 (Mar 2017)

Issues

On asymorphisms of groups

Igor Protasov / Serhii Slobodianiuk
Published Online: 2016-09-14 | DOI: https://doi.org/10.1515/jgth-2016-0038

Abstract

Let G, H be groups and let κ be a cardinal. A bijection f:GH is called an asymorphism if, for any X[G]<κ, Y[H]<κ, there exist X[G]<κ, Y[H]<κ such that for all xG and yH, we have f(Xx)Yf(x), f-1(Yy)Xf-1(y). For a set S, [S]<κ denotes the set {SS:|S|<κ}. Let κ and γ be cardinals such that 0<κγ. We prove that any two Abelian groups of cardinality γ are κ-asymorphic, but the free group of rank γ is not κ-asymorphic to an Abelian group provided that either κ<γ or κ=γ and κ is a singular cardinal. It is known [9] that if γ=κ and κ is regular, then any two groups of cardinality κ are κ-asymorphic.

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About the article


Received: 2016-02-26

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, DOI: https://doi.org/10.1515/jgth-2016-0038.

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