On autocentral kernel of groups

Bahri, Shafigh; Kaheni, Azam; Nasrabadi, Mohammad Mehdi

doi:10.22103/jmmr.2025.24620.1745

On autocentral kernel of groups

Document Type : Research Paper

Authors

Shafigh Bahri

Azam Kaheni

Mohammad Mehdi Nasrabadi

Department of Mathematics, University of Birjand, Birjand, Iran

10.22103/jmmr.2025.24620.1745

Abstract

‎Let $G$ be a group, where $\text{Aut}(G)$ denotes the full automorphisms group of $G$ and $L(G)$ represents the absolute center of $G.$ An automorphism $\alpha \in \text{Aut}(G)$ is called an autocentral automorphism if $g^{-1}\alpha(g) \in L(G),$ for all $g \in G .$ In this paper, we introduce the notion of autocentral kernel subgroup for an arbitrary group $G.$ We then investigate and establish several structural properties of this subgroup, providing insights into its role within the broader framework of group theory.

Keywords

Absolute central automorphisms‎

‎Absolute centre‎

‎Autocentral kernel‎

‎Autonilpotent group

Subjects

Algebra

[1] Arora, H. (2019). Various bounds of group of Autocentral Automorphisms. MathLAB Journal, 3, 1-12. http://purkh.com/index.php/mathlab

[2] Davoudirad, S., Moghaddam, M. R. R., & Rostamyari, M. A. (2016). Some properties of central kernel and central autocommutator subgroups. Journal of Algebra and Applications, 15 (7), 1-7. https://doi.org/10.1142/S0219498816501280

[3] Haimo, F. (1955). Normal automorphisms and their xed points. Trans. Amer. Math. Soc. 78 (1), 150-167. https://doi.org/10.1090/s0002-9947-1955-0067894-0

[4] Hangyang, M., & Guo, X. (2015). The absolute center of nite groups. Journal of Group Theory 18 (8), 887{904. https://doi.org/10.1515/jgth-2015-0015

[5] Hegarty, P. (1994). The absolute centre of a group. J. Algebra 169, 929 { 935. https://doi.org/10.1006/jabr.1994.1318

[6] Moghaddam, M.R.R. (2009). Some properties of autocommutator groups. First Two Days Group Theory Seminar, University of Isfahan, Iran, 12-13, March. https://profdoc.um.ac.ir/articles/a/1010256.pdf

[7] Moghaddam, M. R. R., & Safa, H. (2010). Some properties of autocentral automorphisms of a group. Ric. Mat, 59, 257{264. https://doi.org/10.1007/s11587-010-0085-6

[8] Nasrabadi, M.M., & Kaboutari Farimani, Z. (2015). Absolute central automorphisms that are inner. Indagationes Mathematicae, 26, 137{141. https://doi.org/10.1016/j.indag.2014.08.003

[9] Or , R., & Fouladi, S. (2020). The Absolute Center of p-Groups of Maximal Class. Filomat 34(13), 4483{4487. https://doi.org/10.2298/FIL2013483O

[10] Parvaneh, F. (2015). Some properties of autonilpotent groups. Italian Jornal of Pure and Applied Mathematics, 35 ,1-8.

[11] Robinson, D. J. S. (1996). A Course in the Theory of Groups. 2nd ed. Springer-verlag;New York, Heidelberg, Berlin.
https://link.springer.com/book/10.1007/978-1-4419-8594-1

[12] Soleimani, R. (2022). A Note on Absolute Central Automorphisms of Finite p-Groups. Iranian Journal of Mathematical Sciences and Informatics,17(2 ), 97-108. https://doi.org/10.52547/ijmsi.17.2.97