On 𝐼𝐴-Automorphisms that Fix the Centre Element-Wise

Proceedings Mathematical Sciences

Pradeep K Rai¹ 

[1] School of Mathematics, Harish-Chandra Research Institute, Chhatnag Road, Jhunsi, Allahabad 0, India$$

关键词: 𝐼𝐴-automorphism;    class-preserving automorphism;    isoclinism;    central automorphism.;   

DOI  :  

学科分类：数学（综合）

来源: Indian Academy of Sciences

【 摘 要 】

Let 𝐺 be a group. An automorphism of 𝐺 is called an $IA$-automorphism if it induces the identity mapping on $G/𝛾 2(G)$, where $𝛾 2(G)$ 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 $IA_z(G)=mathrm{Autcent}(G)$.

【 授权许可】

Unknown