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.


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