The Influence of Self-Centralizing Subgroups on the Structure of Finite Groups

DOI: https://doi.org/10.62517/jnse.202517106

Author(s)

Xueqing Qin

Affiliation(s)

Department of Mathematics, Guilin Institute of Information Technology, Guilin, Guangxi, China.

Abstract

In this thesis, we explored the properties of by using prop-erties of self-centralizing subgroups of , and we got some new conclusions of -groups. According to the contents, this paper was divided into two parts.In the first part, we mainly gave out concepts of the -groups and introduced their research background and some research results of predecessors.In the second part, we used the self-centralizing subgroups to research the structure of the -groups. If is an -group then the subgroups and the factor groups of are also -group. In addition, we have also obtained some other new results, which are detailed in Section 2.

Keywords

Self-Centralizing Subgroups; -Subgroups; Nilpotent Groups; -Groups; MSC(2000): 20C15,20D10

References

[1] R.Baer, Situation der Untergruppen und Struktur der Gruppe[J]. Sitz. Heidelerg Akad, 1933, Wiss:12-17. [2] R.Dedekind,Uber Grunppen, Deren Aammtliche Theiler Normaltheiler Sind[J]. Math.Ann, 1897, 48:548-561. [3] D.J.Robinson, A Course in the theory of groups[M]. Heidelberg: Springer-Verlag, 1982. [4] J.Beidleman, H.Heineken, Finite solvable groups whose subnormal subgroups permute with certain classes of subgroups[J]. Journal of Group Theory, 2003, 6:139-158. [5] G.Walls, Trivial intersection groups[J]. Arch.Math, 1979,32:1-4. Y.Hochheim and [6] F.Timmesfeld, A note on TI-subgroups[J]. Arch.Math, 1988, 51:97-103. [7] Zh.Li, J.Hai, Class-preserving coleman automorphisms of finite groups whose second maximal subgroups are TI-Subgroups[J]. Journal of Mathematical Research with Applications, 2013, 33:241-245. [8] C.Delizia, U.Jezernik, P.Moravec and C.Nicotera, Groups in which every non-cyclic subgr-oup contains its centralizer[J]. Algebra Appl, 2014, 13:11-13. [9] C.Delizia, H.Dietrich, P.Moravec and C.Nicotera, Groups in which every non-abelian subgroup contains is self-centralizing[J]. Journal of Algebra, 2016, 462:23-36. [10] M.Hassanzadeh, Z.Mostaghim, On groups whose self-centralizing subgroups are normal[J]. Journal of Algebra And Its Applications, 2019, 1950106(6 pages). [11] Xu Mingyao, An Introduction to Finite Groups (Volume 1) [M]. Beijing: Science Press, 2001.