TY - JOUR
T1 - On Centralizer Subalgebras of Group Algebras
AU - Li , Zhengxing
AU - Hai , Jinke
JO - Communications in Mathematical Research 
VL - 3
SP - 227
EP - 233
PY - 2021
DA - 2021/05
SN - 27
DO - http://doi.org/
UR - https://global-sci.org/intro/article_detail/cmr/19085.html
KW - group ring, centralizer subalgebra, $H$-conjugacy class, $H$-conjugacy
class sum.
AB - <p style="text-align: justify;">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 $\overline{G}&nbsp;= G/N$. 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, $γ(C_{RG}(H)) = C_{R\overline{G}}(\overline{H})$.</p>