Autocommutators and auto-Bell groups
Bull. Korean Math. Soc. 2014 Vol. 51, No. 4, 923-931
Published online July 31, 2014
Mohammad Reza R. Moghaddam, Hesam Safa, and Azam K. Mousavi
Ferdowsi University of Mashhad, University of Bojnord, Ferdowsi University of Mashhad
Abstract : Let $x$ be an element of a group $G$ and $\alpha$ be an automorphism of $G$. Then for a positive integer $n$, the autocommutator $[x,_{n}\alpha]$ is defined inductively by $[x,\alpha]=x^{-1}x^{\alpha}=x^{-1}\alpha(x)$ and $[x,_{n+1}\alpha]=[[x,_{n}\alpha],\alpha]$. We call the group $G$ to be $n$-auto-Engel if $[x,_{n}\alpha]=[\alpha,_{n}x]=1$ for all $x\in G$ and every $\alpha\in {\rm Aut}(G)$, where $[\alpha,x]=[x,\alpha]^{-1}$. Also, for any integer $n\not= 0,1$, a group $G$ is called an $n$-auto-Bell group when $[x^n,\alpha]=[x,{\alpha}^n]$ for every $x\in G$ and each $\alpha\in {\rm Aut}(G)$. In this paper, we investigate the properties of such groups and show that if $G$ is an $n$-auto-Bell group, then the factor group $G/L_3(G)$ has finite exponent dividing $2n(n-1)$, where $L_3(G)$ is the third term of the upper autocentral series of $G$. Also, we give some examples and results about $n$-auto-Bell abelian groups.
Keywords : $n$-auto-Bell group, autocentral series, autocommutator subgroup, $n$-auto-Engel group, $n$-Bell group
MSC numbers : Primary 20D45, 20F12; Secondary 20E36, 20D15
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