Research Article
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Year 2019, , 1808 - 1814, 08.12.2019
https://doi.org/10.15672/HJMS.2018.644

Abstract

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.

Second centralizers and autocommutator subgroups of automorphisms

Year 2019, , 1808 - 1814, 08.12.2019
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. 

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.
There are 12 citations in total.

Details

Primary Language English
Subjects Mathematical Sciences
Journal Section Mathematics
Authors

M. Badrkhani Asl This is me 0000-0003-0359-1523

Mohammad Reza R. Moghaddam This is me 0000-0003-2979-2390

Publication Date December 8, 2019
Published in Issue Year 2019

Cite

APA Asl, M. B., & Moghaddam, M. R. R. (2019). Second centralizers and autocommutator subgroups of automorphisms. Hacettepe Journal of Mathematics and Statistics, 48(6), 1808-1814. https://doi.org/10.15672/HJMS.2018.644
AMA Asl MB, Moghaddam MRR. Second centralizers and autocommutator subgroups of automorphisms. Hacettepe Journal of Mathematics and Statistics. December 2019;48(6):1808-1814. doi:10.15672/HJMS.2018.644
Chicago Asl, M. Badrkhani, and Mohammad Reza R. Moghaddam. “Second Centralizers and Autocommutator Subgroups of Automorphisms”. Hacettepe Journal of Mathematics and Statistics 48, no. 6 (December 2019): 1808-14. https://doi.org/10.15672/HJMS.2018.644.
EndNote Asl MB, Moghaddam MRR (December 1, 2019) Second centralizers and autocommutator subgroups of automorphisms. Hacettepe Journal of Mathematics and Statistics 48 6 1808–1814.
IEEE M. B. Asl and M. R. R. Moghaddam, “Second centralizers and autocommutator subgroups of automorphisms”, Hacettepe Journal of Mathematics and Statistics, vol. 48, no. 6, pp. 1808–1814, 2019, doi: 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.
JAMA Asl MB, Moghaddam MRR. Second centralizers and autocommutator subgroups of automorphisms. Hacettepe Journal of Mathematics and Statistics. 2019;48:1808–1814.
MLA Asl, M. Badrkhani and Mohammad Reza R. Moghaddam. “Second Centralizers and Autocommutator Subgroups of Automorphisms”. Hacettepe Journal of Mathematics and Statistics, vol. 48, no. 6, 2019, pp. 1808-14, doi:10.15672/HJMS.2018.644.
Vancouver Asl MB, Moghaddam MRR. Second centralizers and autocommutator subgroups of automorphisms. Hacettepe Journal of Mathematics and Statistics. 2019;48(6):1808-14.