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12.26 (1992)

Solved

(Shi Shengming). Is it true that a finite $p$-soluble group $G$ has a $p$-block of defect zero if and only if there exists an element $x \in O_{p'}(G)$ such that $C_G(x)$ is a $p'$-subgroup?

Contributor: Jiping Zhang

Progress

No, it is not. The group $3^2.\text{GL}_2(3)$ has a 2-block of defect 0, but the centralizer of every element from $O(G)$ has even order (V. I. Zenkov, in: Trudy Inst. Matem. i Mekh. UrO RAN, 3 (1995), 36–40 (Russian)).

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