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 <TEX>$[x,_n{\alpha}]$</TEX> is defined inductively by <TEX>$[x,{\alpha}]=x^{-1}x^{\alpha}=x^{-1}{\alpha}(x)$</TEX> and <TEX>$[x,_{n+1}{\alpha}]=[[x,_n{\alpha}],{\alpha}]$</TEX>. We call the group G to be n-auto-Engel if <TEX>$[x,_n{\alpha}]=[{\alpha},_nx]=1$</TEX> for all <TEX>$x{\in}G$</TEX> and every <TEX>${\alpha}{\in}Aut(G)$</TEX>, where <TEX>$[{\alpha},x]=[x,{\alpha}]^{-1}$</TEX>. Also, for any integer <TEX>$n{\neq}0$</TEX>, 1, a group G is called an n-auto-Bell group when <TEX>$[x^n,{\alpha}]=[x,{\alpha}^n]$</TEX> for every <TEX>$x{\in}G$</TEX> and each <TEX>${\alpha}{\in}Aut(G)$</TEX>. 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 <TEX>$G/L_3(G)$</TEX> has finite exponent dividing 2n(n-1), where <TEX>$L_3(G)$</TEX> is the third term of the upper autocentral series of G. Also, we give some examples and results about n-auto-Bell abelian groups.