21.54 (2026)

Let $G$ be a finite soluble group with triality, which means that $G$ admits a group of automorphisms $S$ isomorphic to the symmetric group of degree 3 given by the presentation $S = \langle \sigma, \rho \mid \sigma^2 = \rho^3 = 1; \sigma\rho\sigma = \rho^2 \rangle$ such that $m \cdot m^\rho \cdot m^{\rho^2} = 1$ for all $m$ in the set of commutators $M(G) := \{[g, \sigma] \mid g \in G\}$.

Suppose in addition that $G = [G, S]$, the group $G$ is generated by $d$ elements of $M(G)$ and their images under $S$, and $x^n = 1$ for all $x \in M(G)$. Is it true that the Fitting height of $G$ is bounded in terms of $d$ and $n$?

An affirmative answer would provide a reduction of the analogue of the Restricted Burnside Problem for Moufang loops to the nilpotent case.