Second centralizers and autocommutator subgroups of automorphisms

Yıl 2019, Cilt: 48 Sayı: 6, 1808 - 1814, 08.12.2019

M. Badrkhani Asl Mohammad Reza R. Moghaddam

https://doi.org/10.15672/HJMS.2018.644

Öz

In 1994, Hegarty introduced the notion of $K(G)$ and $L(G)$, the autocommutator and autocentral subgroups of $G$, respectively. He proved that if ${G}/{L(G)}$ is finite, then so is $K(G)$ and for the converse he showed that the finiteness of $K(G)$ and $Aut(G)$ gives that ${G}/{L(G)}$ is also finite. In the present article, we construct a precise upper bound for the order of the autocentral factor group ${G}/{L(G)}$, when $K(G)$ is finite and $Aut(G)$ is finitely generated. In 2012, Endimioni and Moravec showed that if the centralizer of an automorphism $\alpha$ of a polycyclic group $G$ is finite, then $L(G)$ and $G/K(G)$ are both finite. Finally, we show that if in a 2-auto-Engel polycyclic group $G$, there exist two automorphisms $\alpha_1$ and $\alpha_2$ such that $C_G(\alpha_1,\alpha_2)=\{g\in G| [g,\alpha_1,\alpha_2]=1\}$ is finite, then $L_2(G)$ and $G/K_2(G)$ are both finite. 

Anahtar Kelimeler

Polycyclic groups, auto-Engel group, autocentral and auocommutator subgroups.

Kaynakça

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