21.50 (2026)

Does every finite 3-group $T$ have a nontrivial characteristic subgroup $C$ such that if $T$ is a Sylow 3-subgroup of a finite group $G$, then $T \cap G' = T \cap H'$, where $H = N_G(C)$?

Such a characteristic subgroup is known to exist in $p$-groups for $p \geqslant 5$ (G. Glauberman, Math. Z., 117 (1970), 46–56), and for $p = 3$ there are two characteristic subgroups $K_1, K_2$ such that $T \cap G' = (T \cap H_1')(S \cap H_2')$, where $H_i = N_G(K_i)$ (G. Glauberman, J. Algebra, 648 (2024), 62–86). The group $S_4$ shows that no such characteristic subgroups can be found in some Sylow 2-subgroups.