Geodesic
On blocks of defect $0$ in finite groups
Izvestiya. Mathematics, Tome 34 (1990) no. 3, pp. 677-683 
S. P. Strunkov. On blocks of defect $0$ in finite groups. Izvestiya. Mathematics, Tome 34 (1990) no. 3, pp. 677-683. http://geodesic.mathdoc.fr/item/IM2_1990_34_3_a7/
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     author = {S. P. Strunkov},
     title = {On blocks of defect~$0$ in finite groups},
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     number = {3},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/IM2_1990_34_3_a7/}
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Voir la notice de l'article provenant de la source Math-Net.Ru

Let $n\geqslant1$ be a given natural number. It is proved that a finite group $G$ has a $p$-block of defect $0$ if and only if for some $g\in G$ the number of solutions of the equation $g=[x_1,x_2]\dots[x_{2n-1},x_{2n}]$ is not divisible by $p$. A number of criteria for the existence of real characters of defect $0$ in $G$ is obtained. Bibliography: 6 titles.

[1] Kertis Ch., Rainer I., Teoriya predstavlenii konechnykh grupp i assotsiativnykh algebr, Nauka, M., 1969 | MR

[2] Isaacs I. M., Character theory of finite groups, Acad. Press, N.Y., 1976 | MR | Zbl

[3] Bryant R. M., Kovács L. G., “A note on generalized characters”, Bull. Austral. Math. Soc., 5:2 (1971), 265–269 | DOI | MR | Zbl

[4] Feit W., The representation theory of finite groups, North Holland, Amsterdam, N.Y., Oxford, 1982 | MR | Zbl

[5] Tsushima Y., “On the blocks of defect zero”, Nagoya Math. J., 44 (1971), 57–59 | MR | Zbl

[6] Tsushima Y., “On the existence of characters of defect zero”, Osaka J. Math., 11 (1974), 417–423 | MR | Zbl