Second centralizers and autocommutator subgroups of automorphisms

Authors : M. Badrkhani ASL, Mohammad Reza R. MOGHADDAM

Pages : 1808-1814

Doi:10.15672/HJMS.2018.644

View : 18 | Download : 10

Publication Date : 2019-12-08

Article Type : Research Paper

Abstract :In 1994, Hegarty introduced the notion of $Kinsert ignore into journalissuearticles values(G);$ and $Linsert ignore into journalissuearticles values(G);$, the autocommutator and autocentral subgroups of $G$, respectively. He proved that if ${G}/{Linsert ignore into journalissuearticles values(G);}$ is finite, then so is $Kinsert ignore into journalissuearticles values(G);$ and for the converse he showed that the finiteness of $Kinsert ignore into journalissuearticles values(G);$ and $Autinsert ignore into journalissuearticles values(G);$ gives that ${G}/{Linsert ignore into journalissuearticles values(G);}$ is also finite. In the present article, we construct a precise upper bound for the order of the autocentral factor group ${G}/{Linsert ignore into journalissuearticles values(G);}$, when $Kinsert ignore into journalissuearticles values(G);$ is finite and $Autinsert ignore into journalissuearticles values(G);$ 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 $Linsert ignore into journalissuearticles values(G);$ and $G/Kinsert ignore into journalissuearticles values(G);$ 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 $C_Ginsert ignore into journalissuearticles values(\alpha_1,\alpha_2);=\{g\in G| [g,\alpha_1,\alpha_2]=1\}$ is finite, then $L_2insert ignore into journalissuearticles values(G);$ and $G/K_2insert ignore into journalissuearticles values(G);$ are both finite. 
Keywords : Polycyclic groups, auto Engel group, autocentral and auocommutator subgroups