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12.80 (1992)

Solved

(K. W. Roggenkamp).
a) Is it true that the number of $p$-blocks of defect 0 of a finite group $G$ is equal to the number of the conjugacy classes of elements $g \in G$ such that the number of solutions of the equation $[x, y] = g$ in $G$ is not divisible by $p$?
b) The same question in the case of $G$ being a simple group.

Contributor: S. P. Strunkov

Progress

a) No it is not true; for example, if $p = 2$ and $G = \mathbb{Z}_3 \times S_3$ (L. Barker, Letter of May, 27, 1996).
b) No, not always. For example in the alternating group $A_5$ for $p = 2$, the number of 2-blocks of defect 0 is 1, but there are 3 conjugacy classes of elements $g$ such that the number of solutions $[x, y] = g$ is odd, namely, $g = (1, 2, 3)$, $(1, 2, 3, 4, 5)$, and $(1, 3, 5, 2, 4)$. (B. Sambale, Letter of 5 May 2021.)

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