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Dergiler / Hacettepe Journal of Mathematics and Statistics / 2019 / Volume: 48 Issue: 6

Second centralizers and autocommutator subgroups of automorphisms

M. Badrkhani Asl(Islamic Azad University)0000-0003-0359-1523
Mohammad Reza R. Moghaddam(Ferdowsi University of Mashhad)0000-0003-2979-2390
Sayfa
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Yayın Tarihi
2019-12-08
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⚡ TL;DR(1 Cümlede Hızlı Çıkarım)

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.

📌 3 Maddede Bu Makale

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Amaç & Hipotez:In 1994, Hegarty introduced the notion of $K(G)$ and $L(G)$, the autocommutator and autocentral subgroups of $G$, respectively.
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Temel Bulgular & Veri: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.
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Sonuç & Katkı: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.
🔬Tür:Research Article
📍Kapsam:2012 Verileri
⚙️Değişkenler:Polycyclic groups · auto-Engel group · autocentral and auocommutator subgroups.

Abstract

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.

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Bu Makaleye Atıf Yap

Asl, M. B., Moghaddam, M. R. R. (2019). Second centralizers and autocommutator subgroups of automorphisms. Hacettepe Journal of Mathematics and Statistics, Volume: 48 Issue: 6. https://doi.org/10.15672/HJMS.2018.644

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Kaynakça (12)

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