Second centralizers and autocommutator subgroups of automorphisms

M. Badrkhani ASL; Mohammad Reza R. MOGHADDAM

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

Makale detayları

Second centralizers and autocommutator subgroups of automorphisms

Yazarlar:

M. Badrkhani ASL | Mohammad Reza R. MOGHADDAM

Yayıncı:

Hacettepe University

Ana Başlıklar:

Matematik

Dergi:

Hacettepe Journal of Mathematics and Statistics 2019 Vol.48 Issue 6, pp.1808-1814

Belge Tipi:

Makale

Yayın Tarihi:

12/2019

Belge Dili:

İngilizce

Document ID (Org No.):

2000451364

Sıklık:

İki Aylık

eISSN:

2651-477X

pISSN:

2651-477X

DOI:

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

Özet:

 

In 1994, Hegarty introduced the notion of K(G)K(G)K(G) and L(G)L(G)L(G), the autocommutator and autocentral subgroups of GGG, respectively. He proved that if G/L(G)G/L(G){G}/{L(G)} is finite, then so is K(G)K(G)K(G) and for the converse he showed that the finiteness of K(G)K(G)K(G) and Aut(G)Aut(G)Aut(G) gives that G/L(G)G/L(G){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)G/L(G){G}/{L(G)}, when K(G)K(G)K(G) is finite and Aut(G)Aut(G)Aut(G) is finitely generated. In 2012, Endimioni and Moravec showed that if the centralizer of an automorphism αα\alpha of a polycyclic group GGG is finite, then L(G)L(G)L(G) and G/K(G)G/K(G)G/K(G) are both finite. Finally, we show that if in a 2-auto-Engel polycyclic group GGG, there exist two automorphisms α1α1\alpha_1 and α2α2\alpha_2 such that CG(α1,α2)={g∈G|[g,α1,α2]=1}CG(α1,α2)={g∈G|[g,α1,α2]=1}C_G(\alpha_1,\alpha_2)=\{g\in G| [g,\alpha_1,\alpha_2]=1\} is finite, then L2(G)L2(G)L_2(G) and G/K2(G)G/K2(G)G/K_2(G) are both finite. .

Makale detayları

Second centralizers and autocommutator subgroups of automorphisms

Yazarlar: M. Badrkhani ASL | Mohammad Reza R. MOGHADDAM

Yayıncı: Hacettepe University

Ana Başlıklar: Matematik

Dergi: Hacettepe Journal of Mathematics and Statistics 2019 Vol.48 Issue 6, pp.1808-1814

Belge Tipi: Makale

Yayın Tarihi: 12/2019

Belge Dili: İngilizce

Document ID (Org No.): 2000451364

Sıklık: İki Aylık

eISSN: 2651-477X

pISSN: 2651-477X

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

Sayı

Yazarlar:

M. Badrkhani ASL | Mohammad Reza R. MOGHADDAM

Yayıncı:

Hacettepe University

Ana Başlıklar:

Matematik

Dergi:

Hacettepe Journal of Mathematics and Statistics 2019 Vol.48 Issue 6, pp.1808-1814

Belge Tipi:

Makale

Yayın Tarihi:

12/2019

Belge Dili:

İngilizce

Document ID (Org No.):

2000451364

Sıklık:

İki Aylık

eISSN:

2651-477X

pISSN:

2651-477X

DOI:

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