| Abstract: | Let $G$ be a finite group, $H ≤ G$ and $R$ be a commutative ring withan identity $1_R$. Let $C_{RG}(H) = { α ∈ RG|αh= hα$ for all $h ∈ H }$, which is calledthe centralizer subalgebra of $H$ in $RG$. Obviously, if $H = G$ then $C_{RG}(H)$ is justthe central subalgebra $Z(RG)$ of $RG$. In this note, we show that the set of all $H$-conjugacy class sums of $G$ forms an $R$-basis of $C_{RG}(H)$. Furthermore, let $N$ be anormal subgroup of $G$ and $γ$ the natural epimorphism from $G$ to $overline{G}= G/N$. Then $γ$ induces an epimorphism from $RG$ to $Roverline{G}$, also denoted by $γ$. We also show that if $R$ is a field of characteristic zero, then $γ$ induces an epimorphism from $C_{RG}(H)$ to $C_{Roverline{G}}(overline{H})$, that is, $γ(C_{RG}(H)) = C_{Roverline{G}}(overline{H})$. |
