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

• # Fulltext

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

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

• # Proceedings – Mathematical Sciences

Current Issue
Volume 129 | Issue 5
November 2019

• # Editorial Note on Continuous Article Publication

Posted on July 25, 2019