Perfect nonlinear functions are widely used in the design and analyses of cryptosystem. Based on the method of algebraic number theory, the properties of preimage distributions of perfect nonlinear functions over finite abelian group are studied. Necessary conditions for the existence of perfect nonlinear functions over finite abelian group are presented, which proves that there are no perfect nonlinear functions for some abelian groups. Finally, the preimage distributions of perfect nonlinear functions over some prime fields are presented.