Kourovka Notebook Problem 21.65

Definitions and mathematical statement

For a finite soluble group $L$, its Fitting height $h(L)$ is the least length $h$ of a normal series $1=L_0\trianglelefteq\cdots\trianglelefteq L_h=L$ having nilpotent factors; set $h(1)=0$. A group is nilpotent when its lower central series terminates at $1$. Solubility means that the derived series, obtained by repeatedly taking commutator subgroups, terminates.

Let $\phi\in\operatorname{Aut}(G)$ for a finite soluble group $G$. Write $m=|\phi|$ for the order of the automorphism and $s=|C_G(\phi)|$, where $C_G(\phi)=\{g\in G:\phi(g)=g\}$. Part (a) asks whether functions $f(m,s)$ and $a(m)$ exist such that every such pair $(G,\phi)$ has a subgroup $H\le G$ satisfying \[ [G:H]\le f(m,s),\qquad h(H)\le a(m). \] Part (b) asks the stronger height bound $h(H)\le b(\Omega(m))$, while still allowing the index bound to depend on $m,s$. Here $\Omega(m)$ is the number of prime factors of $m$ counted with multiplicity, with $\Omega(1)=0$. This is the composition length of the cyclic group $\langle\phi\rangle$.

All bounding functions must work for every group and automorphism with the indicated parameters. The subgroup may depend on both. It is not required to be normal or $\phi$-invariant. Neither question assumes the automorphism order is coprime to $|G|$. In particular, part (b) allows the primes themselves to vary while keeping their total multiplicity fixed.

Short English statement

Does a finite soluble group admit a bounded-index subgroup of Fitting height controlled solely by the order, or the composition length, of a given automorphism?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 2574 (KOU-21.65), 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 21.65. Expanded formulation of the supplied catalogue statement. https://arxiv.org/pdf/1401.0300v47.