AUTOCOMMUTATORS AND AUTO-BELL GROUPS

Title & Authors
AUTOCOMMUTATORS AND AUTO-BELL GROUPS

Abstract
Let x be an element of a group G and be an automorphism of G. Then for a positive integer n, the autocommutator $\small{[x,_n{\alpha}]}$ is defined inductively by $\small{[x,{\alpha}]=x^{-1}x^{\alpha}=x^{-1}{\alpha}(x)}$ and $\small{[x,_{n+1}{\alpha}]=[[x,_n{\alpha}],{\alpha}]}$. We call the group G to be n-auto-Engel if $\small{[x,_n{\alpha}]=[{\alpha},_nx]=1}$ for all $\small{x{\in}G}$ and every $\small{{\alpha}{\in}Aut(G)}$, where $\small{[{\alpha},x]=[x,{\alpha}]^{-1}}$. Also, for any integer $\small{n{\neq}0}$, 1, a group G is called an n-auto-Bell group when $\small{[x^n,{\alpha}]=[x,{\alpha}^n]}$ for every $\small{x{\in}G}$ and each $\small{{\alpha}{\in}Aut(G)}$. 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 $\small{G/L_3(G)}$ has finite exponent dividing 2n(n-1), where $\small{L_3(G)}$ 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;
Language
English
Cited by
1.
Perfect groups and normal subgroups related to an automorphism, Ricerche di Matematica, 2016
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