• On $IA$-Automorphisms that Fix the Centre Element-Wise

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      https://www.ias.ac.in/article/fulltext/pmsc/124/02/0169-0173

    • Keywords

       

      $IA$-automorphism; class-preserving automorphism; isoclinism; central automorphism.

    • Abstract

       

      Let 𝐺 be a group. An automorphism of 𝐺 is called an $IA$-automorphism if it induces the identity mapping on $G/\gamma 2(G)$, where $\gamma 2(G)$ is the commutator sub-group of 𝐺. Let $IA_z(G)$ be the group of those $IA$-automorphisms, which fix the centre element-wise and let Autcent $(G)$ be the group of central automorphisms, the automorphisms that induce the identity mapping on the central quotient. It can be observed that Autcent $(G)=C_{\mathrm{Aut}(G)}(IA_z(G))$. We prove that $IA_z(G)$ and $IA_z(H)$ are isomorphic for any two finite isoclinic groups 𝐺 and 𝐻. Also, for a finite 𝑝-group 𝐺, we give a necessary and sufficient condition to ensure that $IA_z(G)=\mathrm{Autcent}(G)$.

    • Author Affiliations

       

      Pradeep K Rai1

      1. School of Mathematics, Harish-Chandra Research Institute, Chhatnag Road, Jhunsi, Allahabad 211 019, India
    • Dates

       
  • Proceedings – Mathematical Sciences | News

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