5.13 (1976)
SolvedSuppose that $K$ and $L$ are distinct conjugacy classes of involutions in a finite group $G$ and $\langle x, y \rangle$ is a 2-group for all $x \in K$ and $y \in L$. Does it follow that $G \neq [K, L]$?
Progress
Not always; for example, $G = \text{Sp}_4(2^n)$, $n \geqslant 2$, $K, L$ being classes of involutions that have non-trivial intersections with the centres of $N_G(M)'$ and $N_G(N)'$, respectively, where $M, N$ are distinct elementary abelian subgroups of order 8 in a Sylow 2-subgroup of $G$ and the dash means taking the derived subgroup (A. A. Makhnëv, Letter of October, 10, 1981).
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