  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 $G$.
