×

zbMATH — the first resource for mathematics

Some conditions for the existence of \(p\)-blocks of defect 0 in finite groups. (English. Russian original) Zbl 0751.20010
Math. USSR, Sb. 70, No. 2, 587-592 (1991); translation from Mat. Sb. 181, No. 8, 1144-1149 (1990).
Let \(G\) be a finite group, \(S\) a Sylow \(p\)-subgroup of \(G\). A \(p\)-block of \(G\) is called real, if it contains a real absolutely irreducible character. The author proves the following Theorem 1: \(G\) has a \(p\)-block of defect zero if and only if there exists a \(p\)-regular element \(g\in G\) such that \(p| S|^ 2\) does not divide the number of solutions of the equation \(g=x^{-1}uxy^{-1}vy\) \((x,y\in G\); \(u,v\in S\)). Theorem 2: \(G\) has a real \(p\)-block of defect zero if and only if \(p| S|\) does not divide the number of solutions of the equation \(g=x^{-1}uxy^ 2\) (\(x,y\in G\); \(u\in S\)) for some \(p\)-regular element \(g\in G\). Moreover, the author gives constructions of different existence criteria both for \(p\)-blocks of defect zero in \(G\) and for real ones.

MSC:
20C20 Modular representations and characters
20C15 Ordinary representations and characters
20D15 Finite nilpotent groups, \(p\)-groups
20D60 Arithmetic and combinatorial problems involving abstract finite groups
PDF BibTeX XML Cite
Full Text: DOI