 Normalized Fourier analysis on a finite abelian group

For a finite abelian group $G$, use the normalized average ${\mathbb{E}}_x=|G{|}^{-1}{\sum }_x$, normalized convolutionand Fourier transform $\hat{f}(\gamma )={\mathbb{E}}_{x}f(x)\overline{\gamma (x)}$. Then Parseval's identity and the convolution identity take the forms