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Some properties of central kernel quotient of a group

Department of Mathematics, Quchan Branch, Islamic Azad University, Quchan, Iran

Department of Mathematics, Khayyam University, Mashhad, Iran

Department of Pure Mathematics, Center of Excellence in Analysis on Algebraic Structures (CEAAS), Ferdowsi University of Mashhad, P. O. Box 1159, Mashhad 91775, Iran

E-mail Address: rezam@ferdowsi.um.ac.ir

E-mail Address: m.r.moghaddam@khayyam.ac.ir

Corresponding author.

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Mohammad Amin Rostamyari

Department of Mathematics, Khayyam University, Mashhad, Iran

E-mail Address: m.a.rostamyari@khayyam.ac.ir

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https://doi.org/10.1142/S1793557121500479Cited by:0 (Source: Crossref)

Abstract

The authors, in 2016, introduced the notion of central autocommutator subgroup of a group $G$ and proved some new results concerning the central kernel subgroup of $G$ , which was introduced earlier by Haimo in $1955$ . In this paper, we establish some results on central kernel quotient of $G$ . Finally, we introduce the concept of central autonilpotent (henceforth $CA$ -nilpotent) groups and determine its relationship with the nilpotency of the subgroups of central automorphism of $G$ .

Communicated by V. A. Artamonov

Keywords:

AMSC: Primary: 20E45, Primary: 20B30, Secondary: 05A05, Secondary: 05A16

References

1. J. N. S. Bidwell, M. J. Curran and D. J. McCaughan, Automorphisms of direct products of finite groups, Arch. Math. 86 (2006) 481–489. Web of Science, Google Scholar
2. M. J. Curran, Automorphisms of semidirect products, Math. Proc. R. Ir. Acad. A 108 (2008) 205–210. Google Scholar
3. S. Davoudirad, M. R. R. Moghaddam and M. A. Rostamyari, Some properties of central kernel and central autocommutator subgroups, J. Algebra Appl. 15(7) (2016) 1650128, 7 p. Link, Web of Science, Google Scholar
4. S. Davoudirad, M. R. R. Moghaddam and M. A. Rostamyari, Autonilpotent groups and their properties, Asian-Eur. J. Math. 9(2) (2016) 1650056, 7 p. Google Scholar
5. J. Dietz, Automorphisms of products of groups, in Groups St Andrews 2005, Vol. 1 eds. C. M. Campbell et al., London Mathematical Society Lecture Note Series Vol. 339 (CUP, Cambridge 2007), pp. 288–305. Google Scholar
6. F. Haimo, Normal automorphisms and their fixed points, Trans. Amer. Math. Soc. 78(1) (1955) 150–167. Google Scholar
7. P. V. Hegarty, The absolute centre of a group, J. Algebra 169 (1994) 929–935. Web of Science, Google Scholar
8. P. V. Hegarty, Autocommutator subgroups of finite groups, J. Algebra 190 (1997) 556–562. Web of Science, Google Scholar
9. H. Mousavi and A. Shomali, Central automorphisms of semidirect products, Bull. Malays. Math. Sci. Soc. (2) 36(3) (2013) 709–716. Web of Science, Google Scholar
10. D. J. S. Robinson, A Course in the Theory of Groups (Springer-Verlag, New York/Berlin, 1996). Google Scholar
11. M. A. Rostamyari, A. K. Mousavi and M. R. R. Moghaddam, On auto-Engle groups and auto-Bell groups, Southeast Asian Bull. Math. 39 (2015) 265–272. Web of Science, Google Scholar