The kernels of the linear maps of finite Abelian group algebras

TL;DR

This paper characterizes and classifies the kernels of linear maps in finite Abelian group algebras as Mathieu-Zhao spaces, providing necessary and sufficient conditions for their structure.

Contribution

It offers a complete classification of Mathieu-Zhao spaces in finite Abelian group algebras over split fields, based on kernel properties of linear maps.

Findings

Necessary and sufficient conditions for kernels to be Mathieu-Zhao spaces

Complete classification of Mathieu-Zhao spaces in finite Abelian group algebras

Identification of kernel structures over split fields

Abstract

In our paper, we give a necessary and sufficient conditions for the kernels of the linear maps of finite Abelian group algebras to be Mathieu-Zhao spaces of $K[G]$ if $G$ is a finite Abelian group and $K$ is a split field for $G$. Hence we classify all Mathieu-Zhao spaces of the finite Abelian group algebras if $K$ is a split field for $G$.