Year: 2011
Author: Zhengxing Li, Jinke Hai
Communications in Mathematical Research , Vol. 27 (2011), Iss. 3 : pp. 227–233
Abstract
Let $G$ be a finite group, $H ≤ G$ and $R$ be a commutative ring with an identity $1_R$. Let $C_{RG}(H) = \{ α ∈ RG|αh= hα$ for all $h ∈ H \}$, which is called the centralizer subalgebra of $H$ in $RG$. Obviously, if $H = G$ then $C_{RG}(H)$ is just the 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 a normal subgroup of $G$ and $γ$ the natural epimorphism from $G$ to $\overline{G} = G/N$. Then $γ$ induces an epimorphism from $RG$ to $R\overline{G}$, 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 $C_{R\overline{G}}(\overline{H})$, that is, $γ(C_{RG}(H)) = C_{R\overline{G}}(\overline{H})$.
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Journal Article Details
Publisher Name: Global Science Press
Language: English
DOI: https://doi.org/2011-CMR-19085
Communications in Mathematical Research , Vol. 27 (2011), Iss. 3 : pp. 227–233
Published online: 2011-01
AMS Subject Headings: Global Science Press
Copyright: COPYRIGHT: © Global Science Press
Pages: 7
Keywords: group ring centralizer subalgebra $H$-conjugacy class $H$-conjugacy class sum.