Czechoslovak Mathematical Journal

Czechoslovak Mathematical Journal, Vol. 71, No. 4, pp. 1229-1233, 2021

Finite groups in which every self-centralizing subgroup is nilpotent or subnormal or a TI-subgroup

Jiangtao Shi, Na Li

Received November 25, 2020. Published online July 9, 2021.

Abstract: Let $G$ be a finite group. We prove that if every self-centralizing subgroup of $G$ is nilpotent or subnormal or a TI-subgroup, then every subgroup of $G$ is nilpotent or subnormal. Moreover, $G$ has either a normal Sylow $p$-subgroup or a normal $p$-complement for each prime divisor $p$ of $|G|$.

Keywords: self-centralizing; nilpotent; TI-subgroup; subnormal; $p$-complement

Classification MSC: 20D10

DOI: 10.21136/CMJ.2021.0512-20

PDF available at: Springer Institute of Mathematics CAS Digital Mathematics Library

References:
[1] D. J. S. Robinson: A Course in the Theory of Groups. Graduate Texts in Mathematics 80. Springer, New York (1996). DOI 10.1007/978-1-4419-8594-1 | MR 1357169 | Zbl 0836.20001
[2] J. Shi: Finite groups in which every non-abelian subgroup is a TI-subgroup or a subnormal subgroup. J. Algebra Appl. 18 (2019), Article ID 1950159, 4 pages. DOI 10.1142/S0219498819501597 | MR 3977820 | Zbl 07096474
[3] J. Shi, C. Zhang: Finite groups in which every subgroup is a subnormal subgroup or a TI-subgroup. Arch. Math. 101 (2013), 101-104. DOI 10.1007/s00013-013-0545-9 | MR 3089764 | Zbl 1277.20021
[4] J. Shi, C. Zhang: A note on TI-subgroups of a finite group. Algebra Colloq. 21 (2014), 343-346. DOI 10.1142/S1005386714000297 | MR 3192353 | Zbl 1291.20018
[5] Y. Sun, J. Lu, W. Meng: Finite groups whose non-abelian self-centralizing subgroups are TI-subgroups or subnormal subgroups. J. Algebra Appl. 20 (2021), Article ID 2150040, 5 pages. DOI 10.1142/S0219498821500407 | MR 4242212 | Zbl 07347720

Affiliations: Jiangtao Shi (corresponding author), School of Mathematics and Information Sciences, Yantai University, 30, Qingquan RD, Laishan District, Yantai 264005, P. R. China, e-mail: shijt@pku.org.cn; Na Li, Department of Mathematics, Beijing Jiaotong University, No. 3 Shangyuancun, Beijing 100044, P. R. China, e-mail: ln18865550588@163.com

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