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## Second centralizers and autocommutator subgroups of automorphisms

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.
Polycyclic groups, auto-Engel group, autocentral and auocommutator subgroups.
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Primary Language en Mathematics Mathematics Orcid: 0000-0003-0359-1523Author: M. Badrkhani ASL Institution: Islamic Azad UniversityCountry: Iran Orcid: 0000-0003-2979-2390Author: Mohammad Reza R. MOGHADDAM Institution: Ferdowsi University of MashhadCountry: Iran Publication Date : December 8, 2019
 Bibtex @research article { hujms499969, journal = {Hacettepe Journal of Mathematics and Statistics}, issn = {2651-477X}, eissn = {2651-477X}, address = {}, publisher = {Hacettepe University}, year = {2019}, volume = {48}, pages = {1808 - 1814}, doi = {10.15672/HJMS.2018.644}, title = {Second centralizers and autocommutator subgroups of automorphisms}, key = {cite}, author = {ASL, M. Badrkhani and MOGHADDAM, Mohammad Reza R.} } APA ASL, M , MOGHADDAM, M . (2019). Second centralizers and autocommutator subgroups of automorphisms. Hacettepe Journal of Mathematics and Statistics , 48 (6) , 1808-1814 . DOI: 10.15672/HJMS.2018.644 MLA ASL, M , MOGHADDAM, M . "Second centralizers and autocommutator subgroups of automorphisms". Hacettepe Journal of Mathematics and Statistics 48 (2019 ): 1808-1814 Chicago ASL, M , MOGHADDAM, M . "Second centralizers and autocommutator subgroups of automorphisms". Hacettepe Journal of Mathematics and Statistics 48 (2019 ): 1808-1814 RIS TY - JOUR T1 - Second centralizers and autocommutator subgroups of automorphisms AU - M. Badrkhani ASL , Mohammad Reza R. MOGHADDAM Y1 - 2019 PY - 2019 N1 - doi: 10.15672/HJMS.2018.644 DO - 10.15672/HJMS.2018.644 T2 - Hacettepe Journal of Mathematics and Statistics JF - Journal JO - JOR SP - 1808 EP - 1814 VL - 48 IS - 6 SN - 2651-477X-2651-477X M3 - doi: 10.15672/HJMS.2018.644 UR - https://doi.org/10.15672/HJMS.2018.644 Y2 - 2018 ER - EndNote %0 Hacettepe Journal of Mathematics and Statistics Second centralizers and autocommutator subgroups of automorphisms %A M. Badrkhani ASL , Mohammad Reza R. MOGHADDAM %T Second centralizers and autocommutator subgroups of automorphisms %D 2019 %J Hacettepe Journal of Mathematics and Statistics %P 2651-477X-2651-477X %V 48 %N 6 %R doi: 10.15672/HJMS.2018.644 %U 10.15672/HJMS.2018.644 ISNAD ASL, M. Badrkhani , MOGHADDAM, Mohammad Reza R. . "Second centralizers and autocommutator subgroups of automorphisms". Hacettepe Journal of Mathematics and Statistics 48 / 6 (December 2019): 1808-1814 . https://doi.org/10.15672/HJMS.2018.644 AMA ASL M , MOGHADDAM M . Second centralizers and autocommutator subgroups of automorphisms. Hacettepe Journal of Mathematics and Statistics. 2019; 48(6): 1808-1814. Vancouver ASL M , MOGHADDAM M . Second centralizers and autocommutator subgroups of automorphisms. Hacettepe Journal of Mathematics and Statistics. 2019; 48(6): 1814-1808.