Kourovka Problem 20.2 — totally $3$-closed simple groups

Solved with Albilich (AI). Human authors: Ting Gong, Yong Yang, Michael Ruofan Zeng. AI system: Albilich.

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Answered affirmatively. Finite simple groups of Lie type that are totally $3$-closed do exist. the smallest example is $\operatorname{PSL}_2(7)$. Moreover, we prove a strengthened version of the original problem: we determine total $3$-closedness throughout the projective special linear family, apart from one remaining infinite family.

Theorem. Let $G = \operatorname{PSL}_n(q)$ be a finite nonabelian simple group. Then:

1. if $n = 2$, then $G$ is totally $3$-closed if and only if $q \ge 7$ is prime;
2. if $n = 3$, then $G$ is totally $3$-closed if and only if either $q = 3$, or $q$ is prime with $q \equiv 2 \pmod 3$;
3. if $n = 4$, then $G$ is not totally $3$-closed for any $q$;
4. if $n \ge 5$ and $q > 2$, then $G$ is not totally $3$-closed.

This settles the existence half of Problem 20.2 affirmatively. Scope of what remains open: within $\operatorname{PSL}_n(q)$ the only unresolved case is $n \ge 5$ with $q = 2$; simple groups of other Lie types are not treated here, so the "full classification" half of the problem is not yet complete outside this family.

Proof sketch

Simplicity gives an effective reduction from arbitrary faithful actions to coset actions: a finite nonabelian simple $S$ is totally $3$-closed if and only if the diagonal action of $S$ on $S/H \sqcup S/K$ is $3$-closed for every pair of proper subgroups $H, K < S$ (repetition allowed). Any faithful action with a base of size at most two is $3$-closed, and such an action also governs its union with any other transitive $3$-closed action, so the positive cases reduce to the relatively few subgroups whose coset actions have no base of size two.

For $\operatorname{PSL}_2(p)$, Dickson's subgroup classification leaves torus normalisers, exceptional subgroups, and subgroups of a Borel subgroup. The first two are handled by base-size estimates; the Borel subgroups give scalar-fibre actions above $\mathbb{P}^1(\mathbb{F}_p)$, whose $3$-closures are determined by the projective quotient and by determinant classes on the fibres. The remaining small primes are treated using the Fano plane, the unique $2$-$(11,5,2)$ biplane, Paley graphs and tournaments, and the Perkel graph.

$\operatorname{PSL}_3(p)$ follows a similar structure. Nonparabolic maximal subgroups use the subgroup classification together with explicit trivial-intersection conjugates; subgroups of point and line parabolics need a separate argument, where mixed triple orbits recover equality or incidence in $\operatorname{PG}(2,p)$ and force the actions on different projective fibres to come from a single group element. The case $\operatorname{PSL}_3(3)$ is settled by one exhaustive GAP computation, reproduced with its output in the appendix.

The negative results come from local matching. On projective points $\operatorname{PGL}_n(q)$ and $\operatorname{PSL}_n(q)$ have the same orbits on ordered triples, and field automorphisms preserve these orbits, giving a semilinear obstruction whenever $q$ is a proper prime power or $\gcd(n, q-1) > 1$. For $n \ge 4$ and $q > 2$ one uses the action on nonzero vectors modulo the centre of $\operatorname{SL}_n(q)$: every element of the corresponding general linear quotient can be matched on any ordered $k$-tuple with $k < n$ by an element of the special linear quotient. The exceptional isomorphism $\operatorname{PSL}_4(2) \cong A_8$ gives the last negative case in dimension four.

Reference

Ting Gong, Yong Yang, Michael Ruofan Zeng, Total 3-closure for projective special linear groups, arXiv:2608.17878 (submitted 18 August 2026, 26 pp.). Theorem 1.1; MSC 20B25, 20D06.

Attribution

Human authors: Ting Gong (University of Washington), Yong Yang (Texas State University), Michael Ruofan Zeng (University of Washington).

AI system: Albilich — an open-source agentic proof-state harness developed by the authors. Albilich first certified the explicit example $\operatorname{PSL}_2(7)$, reducing arbitrary faithful actions to unions of at most two transitive coset actions, verifying the resulting finite configurations with GAP, and passing back to arbitrary faithful actions by a synchronisation argument; further Albilich-assisted runs produced the family-level classification above. All statements and proofs were checked and revised by the human authors, who take full responsibility for the contents. Albilich is described in Ting Gong, Michael Ruofan Zeng, Yong Yang, Albilich: Steerable Proof-State Orchestration for LLM-Based Mathematical Research with CAS Integration, arXiv:2607.27705.