Bulletin of the Korean Mathematical Society (대한수학회보)

Korean Mathematical Society (대한수학회)
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AUTOCOMMUTATORS AND AUTO-BELL GROUPS

  • Moghaddam, Mohammad Reza R. (Department of Mathematics Khayyam Higher Education Institute, Centre of Excellence in Analysis on Algebraic Structures Ferdowsi University of Mashhad) ;
  • Safa, Hesam (Department of Mathematics Faculty of Basic Sciences University of Bojnord) ;
  • Mousavi, Azam K. (Faculty of Mathematical Sciences International Branch Ferdowsi University of Mashhad, Centre of Excellence in Analysis on Algebraic Structures Ferdowsi University of Mashhad)
  • Received : 2012.05.21
  • Published : 2014.07.31
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Abstract

Let x be an element of a group G and 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 $[x,_n{\alpha}]=[{\alpha},_nx]=1$ for all $x{\in}G$ and every ${\alpha}{\in}Aut(G)$, where $[{\alpha},x]=[x,{\alpha}]^{-1}$. Also, for any integer $n{\neq}0$, 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 ${\alpha}{\in}Aut(G)$. 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.

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References

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