Irreducible Characters with Cyclic Anchor Group

We consider G to be a finite group and p as a prime number. We fix psi to be an irreducible character of G with its restriction to all p-regular elements of G and psi 0 to be an irreducible Brauer character. The main aim of this paper is to describe and investigate the relationship between cyclic anchor group of psi and the defect group of a p-block which contains psi. Our methods are to study and generalize some facts for the cyclic defect groups of a p-block B to the case of a cyclic anchor group of irreducible characters which belong to B. We establish and prove a criteria for an irreducible character to have a cyclic anchor group.