Kourovka Notebook Problem 17.119

Definitions and mathematical statement

For a finite group $H$, its rank $\operatorname{rk}(H)$ is the least nonnegative integer $s$ such that every subgroup of $H$ can be generated by at most $s$ elements. A group is soluble if its derived series terminates at the trivial subgroup. Let $F(H)$ denote the Fitting subgroup of a finite group $H$, namely its largest nilpotent normal subgroup. Define the upper Fitting series by \[ F_0(H)=1,\qquad F_{i+1}(H)/F_i(H)=F\left(H/F_i(H)\right)\quad(i\geq 0). \] Let $G$ be a finite soluble group and let $A\leq\operatorname{Aut}(G)$ be a finite soluble group with $\gcd(|A|,|G|)=1$. Set \[ C_G(A)=\{g\in G:\alpha(g)=g\text{ for every }\alpha\in A\},\qquad r=\operatorname{rk}(C_G(A)). \] Write $|A|=p_1\cdots p_l$, where the primes are counted with multiplicity; for $|A|=1$, use the empty product and $l=0$.

The question is whether there is a universal function $f:\mathbb Z_{\geq0}\to\mathbb Z_{\geq0}$ growing at most linearly, together with a function \[ B:\mathbb Z_{\geq1}\times\mathbb Z_{\geq0}\longrightarrow\mathbb Z_{\geq0}, \] such that, for every such pair $(G,A)$, \[ \operatorname{rk}\left(G/F_{f(l)}(G)\right)\leq B(|A|,r). \] Here growing at most linearly means that there are universal nonnegative constants $c,d$ such that $f(l)\leq cl+d$ for every $l\geq0$. Thus $f$ is independent of $G$, $A$, and $r$, and the rank bound depends only on $|A|$ and $r$. The source says "linear function" without specifying whether an additive constant is allowed; the formulation above records the at-most-linear growth interpretation.

The supplied source also reports that an exponential function $f$ with the required rank-bound property was obtained in the cited Khukhro--Mazurov reference. It further attributes to Hartley--Isaacs the assertion that there is a function $D$, depending only on its displayed arguments, such that \[ \left|G/F_{2l+1}(G)\right|\leq D\left(|C_G(A)|,|A|\right) \] under the same hypotheses. These are supplied background assertions, not independently verified results in this draft.

Short English statement

Does a universal number of Fitting-series steps growing linearly with the number of prime factors of $|A|$, counted with multiplicity, leave a quotient whose rank is bounded only by $|A|$ and the rank of the common fixed-point subgroup? The hypotheses require both groups to be finite and soluble and their orders to be coprime. The registry labels the problem open; this draft does not independently verify that status.

Sources

• Local UnsolvedMath registry: permanent ID $701420$, exact source code catalogue entry, title "Kourovka Notebook Problem 17.119." The supplied record labels the problem open and attributes it to E. I. Khukhro, proposed in 2010.
• The Kourovka Notebook (Unsolved Problems in Group Theory), original problem 17.119, 17th issue (2010); the supplied background also identifies the 21st-issue edition (2026), arXiv:1401.0300v48. The supplied collection description names E. I. Khukhro and V. D. Mazurov as editors.
• Supplied reference for the exponential-index rank bound: E. Khukhro and V. Mazurov, Groups St. Andrews 2005, vol. II, Cambridge Univ. Press, 2007, pp. 564--585. The record does not give an article title.
• Supplied attribution for the bound on $|G/F_{2l+1}(G)|$: Hartley--Isaacs. No bibliographic details for this result are supplied.