TY  - JOUR
T1  - Second centralizers and autocommutator subgroups of automorphisms
AU  - Asl, M. Badrkhani
AU  - Moghaddam, Mohammad Reza R.
PY  - 2019
DA  - December
DO  - 10.15672/HJMS.2018.644
JF  - Hacettepe Journal of Mathematics and Statistics
PB  - Hacettepe University
WT  - DergiPark
SN  - 2651-477X
SP  - 1808
EP  - 1814
VL  - 48
IS  - 6
LA  - en
AB  - 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.
KW  - Polycyclic groups
KW  - auto-Engel group
KW  - autocentral and auocommutator subgroups.
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UR  - https://doi.org/10.15672/HJMS.2018.644
L1  - https://dergipark.org.tr/en/download/article-file/601991
ER  -