## Existence and number of $$p$$-blocks of defect 0 in finite groups.(Russian)Zbl 0766.20004

Let $$G$$ be a finite group, $$p$$ be a prime number dividing its order $$| G|$$, $$\chi$$ be the character of some ordinary, absolutely irreducible representation of the group $$G$$. If $$| G|=p^ em$$, $$\chi(1)=p^ cq$$ $$(m$$ and $$q$$ are relatively prime to $$p)$$, then the nonnegative number $$e-c$$ is called the $$p$$-defect of the character $$\chi$$. (For characters of $$p$$-defect 0, the corresponding ideals of the group algebra are its $$p$$-blocks.) In the present paper, a construction is given, which allows one to obtain infinitely many criteria of the existence of irreducible, complex and real characters of $$p$$-defect 0, characterizations of the number of these characters in arithmetical terms are presented. The methods applied allow one to obtain answers to the analogous questions for characters of minimal $$p$$-defect.

### MSC:

 20C20 Modular representations and characters 20C15 Ordinary representations and characters 20D60 Arithmetic and combinatorial problems involving abstract finite groups
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