Kourovka Notebook Problem 21.82

Definitions and mathematical statement

Let $F_n=\langle x_1,\ldots,x_n\rangle$ denote the free group on $n$ generators, where $n$ is an integer at least three. A subgroup $N\le F_n$ is characteristic if $\alpha(N)=N$ for every automorphism $\alpha:F_n\to F_n$. Characteristic subgroups are normal, since conjugations are automorphisms, so the quotient $F_n/N$ is defined.

A characteristic quotient of $F_n$ means a quotient by a characteristic subgroup. A finite simple group means a nontrivial finite group with no proper nontrivial normal subgroup; this convention includes cyclic groups of prime order. The conjecture asserts \[ \nexists N\le F_n:\quad N\text{ characteristic in }F_n \quad\text{and}\quad F_n/N\text{ finite simple}. \] Equivalently, there should be no surjective homomorphism from $F_n$ to any finite simple group whose kernel is preserved by every automorphism of $F_n$.

A characteristic kernel would allow each automorphism of the free group to induce an automorphism of the finite quotient, compatibly with the quotient map. This is the invariance requirement being tested; mere existence of a finite simple quotient does not impose it. The same assertion is requested separately for each rank $n\ge3$, with no bound on the quotient's size. The trivial quotient is excluded by the definition of simplicity.

Short English statement

Can a free group of rank at least three have a nontrivial finite simple quotient whose kernel is invariant under every automorphism?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 2591 (KOU-21.82), 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, version 47 (30 September 2026), new problems, Problem 21.82. Expanded formulation of the supplied catalogue statement. https://arxiv.org/pdf/1401.0300v47.