Autocommutators and auto-Bell groups. (Q2877794)

Let \(G\) be a group and let \(\Aut(G)\) denote the automorphism group of \(G\). For an integer \(n\neq 0,1\), the group \(G\) is called \(n\)-auto-Bell group if \([x^n,\alpha]=[x,\alpha^n]\) for all \(x\in G\) and all \(\alpha\in\Aut(G)\), where for an automorphism \(\beta\in\Aut(G)\), \([a,\beta]:=a^{-1}a^\beta\) for all \(a\in G\).NEWLINENEWLINE Let \(AR_2(G)=\{g\in G\mid [[g,\alpha],\alpha]=1\text{ for all }\alpha\in\Aut(G)\}\). It is proved [in \textit{M. R. R. Moghaddam, M. Farrokhi} and \textit{H. Safa}, Some properties of 2-auto-Engel groups (submitted)] that \(AR_2(G)\) is a characteristic subgroup of \(G\). For any integer \(n\neq 0,1\), the authors call a group \(G\) an \(n\)-auto-Kappe group if the factor group \(G/AR_2(G)\) has finite exponent dividing \(n\).NEWLINENEWLINE The main results of the paper under review are the following:NEWLINENEWLINE Theorem 2.4. Every \(n\)-auto-Bell group is also \(n(n-1)\)-auto-Kappe and hence \(n(n-1)\)-auto-Bell.NEWLINENEWLINE Theorem 3.1. Let \(G\) be an \(n\)-auto-Bell group. Then the factor group \(G/L_3(G)\) has finite exponent \(2n(n-1)\), where \(L_3(G)\) is the third term of the upper autocentral series of \(G\) which is defined as \(L_3(G)=\{x\in G\mid [[[x,\alpha_1],\alpha_2],\alpha_3]=1\text{ for all }\alpha_i\in\Aut(G)\}\).NEWLINENEWLINE Theorem 4.3. Let \(G\) be a finite Abelian \(n\)-auto-Bell group with \(|G|=\prod_{i=1}^m p_i^{r_i}\). Then for every \(1\leq j\leq m\), the numbers \(p_j(p_j-1)\) and \(\prod_{i=1}^m p_i\) divide \(n\) or \(n-1\) when \(n\) is an even or an odd integer, respectively.NEWLINENEWLINE Reviewer's remark: In the statement of Theorem 4.3, it is not stated that the \(p_i\) are distinct prime numbers.