A Note on 2-Nilpotence of Finite Groups

Proceedings Mathematical Sciences

Cui Zhang¹  Jiangtao Shi² 

[1] University of Primorska, IAM, Muzejski trg , 000 Koper, Slovenia$$;School of Mathematics and Information Science, Yantai University, Yantai 00, China$$

关键词: Finite group;    2-nilpotent group;    quaternion-free.;   

DOI  :  

学科分类：数学（综合）

来源: Indian Academy of Sciences

【 摘 要 】

Ballester-Bolinches and Guo showed that a finite group 𝐺 is 2-nilpotent if 𝐺 satisfies: (1) a Sylow 2-subgroup 𝑃 of 𝐺 is quaternion-free and (2) $𝛺_1(Pcap G')≤ Z(P)$ and $N_G(P)$ is 2-nilpotent. In this paper, it is obtained that 𝐺 is a non-2-nilpotent group of order 16𝑞 for an odd prime 𝑞 satisfying (1) a Sylow 2-subgroup 𝑃 of 𝐺 is not quaternion-free and (2) $𝛺_1(Pcap G')≤ Z(P)$ and $N_G(P)$ is 2-nilpotent if and only if 𝑞=3 and $G≅ GL_2(3)$.

【 授权许可】

Unknown