Houston Journal of Mathematics, ( ISI ), Volume (44), No (1), Year (2018-3) , Pages (31-48)

Title : ( Some properties of 2-auto-Engel groups )

Authors: Hesam Safa , Mohammad Farrokhi D. G , Mohammad Reza Rajabzadeh Moghaddam ,

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Abstract

ABSTRACT. For a given group G, with an element x in G and automorphism α in Aut(G), the nth autocommutator [x,nα] is defined recursively by [x,α]=x-1xα and [x,nα]=[[x,n- 1α],α] for all n>1. The group G is said to be n-auto-Engel if [x,nα]=[α,nx]=1, for all x∈G and all α∈Aut(G), where [α,x]=[x,α]-1. We study the structure of 2-auto-Engel groups and show that 2-auto-Engel groups indeed satisfy the equation α(x)α-1(x)=x2, for all x∈G and α∈Aut (G). Also, we give a precise description of all abelian 2-auto-Engel groups of finite 2-rank as well as 2-auto-Engel 2-groups, which are not purely non-abelian, and construct an infinite family of purely non-abelian 2-auto-Engel 2-groups.

Keywords

, Autocommutator subgroup, Engel element, auto-Engel element, 2-auto-Engel group, autocentral series.
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@article{paperid:1068605,
author = { and and Rajabzadeh Moghaddam, Mohammad Reza},
title = {Some properties of 2-auto-Engel groups},
journal = {Houston Journal of Mathematics},
year = {2018},
volume = {44},
number = {1},
month = {March},
issn = {0362-1588},
pages = {31--48},
numpages = {17},
keywords = {Autocommutator subgroup; Engel element; auto-Engel element; 2-auto-Engel group; autocentral series.},
}

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%0 Journal Article
%T Some properties of 2-auto-Engel groups
%A
%A
%A Rajabzadeh Moghaddam, Mohammad Reza
%J Houston Journal of Mathematics
%@ 0362-1588
%D 2018

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