Second centralizers and autocommutator subgroups of automorphisms

Year 2019, Volume: 48 Issue: 6, 1808 - 1814, 08.12.2019

M. Badrkhani Asl Mohammad Reza R. Moghaddam

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

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. 

Keywords

Polycyclic groups, auto-Engel group, autocentral and auocommutator subgroups.

References

• [1] G. Endimioni and P. Moravec, On the centralizer and the commutator subgroup of an automorphism, Monatshefte für Mathematik, 167, 165–174, 2012.
• [2] T.A. Fournelle, Elementary abelian p-groups as automorphisms groups of infinite groups II, Houston J. Math. 9, 269–276, 1983.
• [3] D. Gumber, H. Kalra and S. Single, Automorphisms of groups and converse of Schur’s theorem, at: http://arXiv.org/math/arXiv:1303.4966v1.
• [4] P.V. Hegarty, The absolute centre of a group, J. Algebra, 169, 929–935, 1994.
• [5] P.V. Hegarty, Autocommutator subgroups of finite groups, J. Algebra, 190, 556–562, 1997.
• [6] M.R.R. Moghaddam, F. Parvaneh and M. Naghshineh, The lower autocentral series of abelian groups, Bull. Korean Math. Soc. 48, 79–83, 2011.
• [7] B.H. Neumann, Groups with finite classes of conjugate elements, Proc. London Math. Soc. 3 (1), 178–187, 1951.
• [8] P. Niroomand, The converse of Schur’s theorem, Arch. Math. 94, 401–403, 2010.
• [9] K. Podoski and B. Szegedy, Bounds for the index of the centre in capable groups, Proc. Amer. Math. Soc. 133, 3441–3445, 2005.
• [10] H. Safa, M. Farrokhi D.G. and M.R.R. Moghaddam, Some properties of 2-auto-Engel groups, Houston J. Math. 44 (1), 31–48, 2018.
• [11] I. Schur, Uber die darstellung der endlichen grouppen durch gebrochene lineare substiutionen, J. Reine Angew. Math. 127, 20–50, 1904.
• [12] B. Sury, A generalization of a converse to Schur’s theorem, Arch. Math. 95, 317–318, 2010.