Kourovka Notebook Problem 21.56

Definitions and mathematical statement

Let $A$ be a finite nilpotent group acting by automorphisms on a finite soluble group $G$. Nilpotent means that the lower central series terminates, and soluble means the same for the derived series. For a finite group $X$, let $\ell(X)$ be its composition length, the number of simple factors in a composition series, with $\ell(1)=0$.

An $A$-composition series of $G$ is an $A$-invariant subnormal series \[ 1=G_0\trianglelefteq G_1\trianglelefteq\cdots \trianglelefteq G_t=G \] with no proper refinement by $A$-invariant subgroups. Its factors are elementary abelian groups viewed as irreducible modules for $A$ over the corresponding prime fields. Call a factor trivial as an $A$-module if every member of $A$ acts as the identity on it. Let $c(G,A)$ count these trivial factors, with repetitions counted, in the specified series.

The Fitting height $h(G)$ is the least length of a normal series of $G$ with nilpotent factors; set $h(1)=0$. The conjecture asks for absolute constants $C_1,C_2\ge0$ such that \[ h(G)\le C_1\ell(A)+C_2c(G,A) \] for every such group action. The constants may not depend on the groups, their orders, the primes involved, or the action.

The action is not assumed coprime, meaning $|A|$ and $|G|$ may share prime factors, and faithfulness is not required. The count concerns trivial composition factors of the action, rather than the number of fixed elements in $G$ or the total number of composition factors.

Short English statement

Can Fitting height under a nilpotent automorphism action be bounded linearly by the acting group’s composition length and the number of trivial action factors?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 2565 (KOU-21.56), Kourovka Notebook - New Problems, Issue 21. Catalogue identity and source transcription; CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 The Kourovka Notebook, 21st edition, new problems, Problem 21.56. Expanded formulation of the supplied catalogue statement. https://arxiv.org/pdf/1401.0300v47.