Kourovka Notebook Problem 21.66

Definitions and mathematical statement

A finite group is soluble if its derived series reaches $1$, where the next term is the subgroup generated by all commutators in the previous term. For a finite soluble group $G$, the Fitting height $h(G)$ is the least number of factors in a normal series from $1$ to $G$ with nilpotent factors, and $h(1)=0$. A group $L$ is nilpotent if its lower central series $\gamma_1(L)=L$, $\gamma_{i+1}(L)=[\gamma_i(L),L]$ reaches $1$.

Let $A\le\operatorname{Aut}(G)$ be a nilpotent group of automorphisms of a finite soluble group $G$. Since $G$ is finite, $A$ is finite. Its common fixed-point subgroup is \[ C_G(A)=\{g\in G:a(g)=g\text{ for every }a\in A\} =\bigcap_{a\in A}C_G(a). \] Is there a function $f:\mathbb N\times\mathbb N\to\mathbb N$ such that \[ h(G)\le f\left(|A|,|C_G(A)|\right) \] for every such pair $(G,A)$?

The same function must control all ambient groups with the same orders of the automorphism group and its common fixed subgroup. There is no assumed bound on $|G|$, and no coprimality condition between $|A|$ and $|G|$. The parameter counts elements fixed by every automorphism in $A$ simultaneously, not a sum of individual fixed-point counts. The conclusion bounds the height of $G$ itself, rather than only the height of some subgroup of bounded index.

Short English statement

Is the Fitting height of a finite soluble group bounded by the order of a nilpotent automorphism group and the number of its common fixed points?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 2575 (KOU-21.66), 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 21.66. Expanded formulation of the supplied catalogue statement. https://arxiv.org/pdf/1401.0300v47.