The applications of probability groups on Hopf algebras

In this work, we use probability groups, introduced by Harrison in 1979, as a tool to study a semisimple Hopf algebra $H$ with a commutative character ring and prove that the algebra generalized by the dual probability group is the center $Z(H)$ of $H$ and the product of two class sums is an integral combination up to a factor of $\dim (H)^{-1}$ of the class sums of $H$. We classify all the 2-integral probability groups with 2 or 3 elements.