Bulletin of the
Korean Mathematical Society BKMS

Let $G$ be a finite group, $K$ a split field for $G$, and $L$ a linear map from $K[G]$ to $K$. In our paper, we first give sufficient and necessary conditions for $\operatorname{Ker}L$ and $\operatorname{Ker}L\cap Z(K[G])$, respectively, to be Mathieu-Zhao spaces for some linear maps $L$. Then we give equivalent conditions for $\operatorname{Ker}L$ to be Mathieu-Zhao spaces of $K[G]$ in term of the degrees of irreducible representations of $G$ over $K$ if $G$ is a finite Abelian group or $G$ has a normal Sylow $p$-subgroup $H$ and $L$ are class functions of $G/H$. In particular, we classify all Mathieu-Zhao spaces of the finite Abelian group algebras if $K$ is a split field for $G$.

Keywords: Mathieu-Zhao spaces, irreducible representations, class functions

MSC numbers: 20C05, 20C15, 20C20, 16D60

Supported by: The author is supported by the NSF of Hunan Province (Grant No. 2023JJ30386), the NSF of China (Grant No. 12371020), the Scientific Research Fund of Hunan Provincial Education Department (Grant No. 21A0056), the China Scholarship Council and the Construct Program of the Key Discipline in Hunan Province.