Bulletin of the
Korean Mathematical Society
BKMS

ISSN(Print) 1015-8634 ISSN(Online) 2234-3016

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Bull. Korean Math. Soc. 2015; 52(4): 1097-1105

Printed July 31, 2015

https://doi.org/10.4134/BKMS.2015.52.4.1097

Copyright © The Korean Mathematical Society.

Non-abelian tensor analogues of 2-auto Engel groups

Mohammad Reza R. Moghaddam and Mohammad Javad Sadeghifard

Islamic Azad University, Islamic Azad University

Abstract

\noindent The concept of tensor analogues of right 2-Engel elements in groups were defined and studied by Biddle and Kappe \cite{dp} and Moravec \cite{pm}. Using the automorphisms of a given group $G$, we introduce the notion of tensor analogue of 2-auto Engel elements in $G$ and investigate their properties. Also the concept of $2_{\otimes}$-auto Engel groups is introduced and we prove that if $G$ is a $2_{\otimes}$-auto Engel group, then $G\otimes {\rm Aut}(G)$ is abelian. Finally, we construct a non-abelian $2$-auto-Engel group $G$ so that its non-abelian tensor product by ${\rm Aut}(G)$ is abelian.

Keywords: non-abelian tensor product, auto-Engel element, autocommutator subgroup, absolute centre

MSC numbers: Primary 20F28, 20F45; Secondary 20F99

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