Research Article
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Year 2019, Volume 48, Issue 6, 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, Volume 48, Issue 6, 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.

Details

Primary Language English
Subjects Mathematics
Journal Section Mathematics
Authors

M. Badrkhani ASL This is me
Islamic Azad University
0000-0003-0359-1523
Iran


Mohammad Reza R. MOGHADDAM This is me
Ferdowsi University of Mashhad
0000-0003-2979-2390
Iran

Publication Date December 8, 2019
Published in Issue Year 2019, Volume 48, Issue 6

Cite

Bibtex @research article { hujms499969, journal = {Hacettepe Journal of Mathematics and Statistics}, issn = {2651-477X}, eissn = {2651-477X}, address = {}, publisher = {Hacettepe University}, year = {2019}, 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. B. & Moghaddam, M. R. R. (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. B. , Moghaddam, M. R. R. "Second centralizers and autocommutator subgroups of automorphisms" . Hacettepe Journal of Mathematics and Statistics 48 (2019 ): 1808-1814 <https://dergipark.org.tr/en/pub/hujms/article/499969>
Chicago Asl, M. B. , Moghaddam, M. R. R. "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. B. , Moghaddam M. R. R. Second centralizers and autocommutator subgroups of automorphisms. Hacettepe Journal of Mathematics and Statistics. 2019; 48(6): 1808-1814.
Vancouver Asl M. B. , Moghaddam M. R. R. Second centralizers and autocommutator subgroups of automorphisms. Hacettepe Journal of Mathematics and Statistics. 2019; 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, Dec. 2019, doi:10.15672/HJMS.2018.644