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      Permanent link:
      https://www.ias.ac.in/article/fulltext/pmsc/123/02/0235-0238

    • Keywords

       

      Finite group; 2-nilpotent group; quaternion-free.

    • Abstract

       

      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) $\Omega_1(P\cap G')\leq Z(P)$ 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) $\Omega_1(P\cap G')\leq Z(P)$ and $N_G(P)$ is 2-nilpotent if and only if $q=3$ and $G\cong GL_2(3)$.

    • Author Affiliations

       

      Jiangtao Shi1 Cui Zhang1 2

      1. School of Mathematics and Information Science, Yantai University, Yantai 264005, China
      2. University of Primorska, IAM, Muzejski trg 2, 6000 Koper, Slovenia

    • Dates

       
      Published
      May 2013
      Manuscript received
      1 July 2012

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