- Isaacs, $\textit{
- Character Theory of Finite Groups
- }$, Theorem(2.4)
- Let $\mathcal{
- K
- }_1,~\mathcal{
- K
- }_2,\cdots,~\mathcal{
- K
- }_r$ be the conjugacy classes of a group $G$. Let $K_i=\sum_{
- x\in\mathcal{
- K
- }_i
- }x\in\mathbb{
- C
- }[G]$. Then the $K_i$ form a basis for $\mathbf{
- Z
- }(\mathbb{
- C
- }[G])$ and if $K_iK_j=\sum a_{
- ijv
- }K_v$, then the multiplication constants $a_{
- ijv
- }$ are nonnegative integers.
- Pf:
- If $z=\sum a_gg\in\mathbf{
- Z
- }(\mathbb{
- C
- }[G])$ and $h\in G$, we have $z=h^{
- -1
- }zh=\sum a_gg^h=\sum a_gg$.
- Pick $g\in\mathcal{
- K
- }_v$. Then $a_{
- ijv
- }$ is the coefficient of $g$ in $K_iK_j$. That is $|\{
- (x,y)|x\in\mathcal{
- K
- }_i,y\in\mathcal{
- K
- }_j,xy=g\
- }|$.
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