21.56 (2026)

Let $\ell(X)$ denote the composition length of a finite group $X$. Let $A$ be a finite nilpotent group acting by automorphisms on a finite soluble group $G$. Let $c(G, A)$ be the number of trivial $A$-modules in a given $A$-composition series of $G$. (Note that $c(G, A) = \ell(C_G(A))$ if $(|A|, |G|) = 1$.)

Conjecture: there are absolute constants $C_1$ and $C_2$ such that the Fitting height of $G$ is at most $C_1\ell(A) + C_2c(G, A)$.