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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
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