Assume that  $G$  is a finite group. Let $\phi=X+iY: G\to S^1$ be a complex function of unit norm.
We consider $G$ as a sample (probability)  space  with the normalized counting measure  $\mu(A)=\frac{n(A)}{|G|}$.  So we consider  $X,Y$  as two random variables. Their covariance is denoted by $Cov(X,Y)$.

>Is the following set $H$ a group with the operation of pointwise multiplication $\phi \psi(x)=\phi(x).\psi(x)$?

$$H=\bigm\{\phi=X+iY:G\to S^1  \bigm| ( \phi\equiv1) \vee  \bigm(\sum_g \phi(g)=0 \wedge  Cov(X,Y)=0 \bigm ) \bigm \}$$


**Remark 1:** Evry character $\phi:G\to S^1=\mathbb{T}$ obviousely belongs to $H$. So this question is an attempt to find a generalization of characters in group theory in the context of probability and statistics.

**Remark 2:** What about if we remove the condition $\sum_g \phi(g)=0$ from the definition of the above $H$?