• Pradeep K Rai

      Articles written in Proceedings – Mathematical Sciences

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

      Pradeep K Rai

      More Details Abstract Fulltext PDF

      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 $IA_z(G)=\mathrm{Autcent}(G)$.

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