Total 3-closure for projective special linear groups

Abstract: A finite group is totally $3$-closed if every faithful permutation representation of it is $3$-closed. We study this property for the finite simple projective special linear groups. We prove that $\PSL_2(q)$ is totally $3$-closed if and only if $q\geq 7$ is prime, and that $\PSL_3(q)$ is totally $3$-closed if and only if either $q=3$, or $q$ is prime and $q\equiv 2\pmod 3$. We further prove that $\PSL_4(q)$ is never totally $3$-closed and that $\PSL_n(q)$ is not totally $3$-closed whenever $n\geq 5$ and $q>2$. Within the family $\PSL_n(q)$, only the groups $\PSL_n(2)$ with $n\geq 5$ remain unresolved. In particular, this answers Problem~20.2 of the Kourovka Notebook affirmatively.

• The paper classifies total 3-closedness for PSL_n(q), proving that PSL_2(q) qualifies exactly when q is prime and q ≥ 7, while PSL_3(q) qualifies for q = 3 or prime q ≡ 2 mod 3.
• The authors reduce the problem to checking diagonal unions of pairs of coset actions, using base-size arguments, subgroup classifications, projective geometry, rigidity methods, and limited computer verification.
• The results show PSL_4(q) and PSL_n(q) for n ≥ 5 and q > 2 are not totally 3-closed, leaving only PSL_n(2) for n ≥ 5 unresolved and answering Kourovka Problem 20.2 affirmatively.

The paper classifies total 3-closedness for finite simple groups $\mathrm{PSL}_n(q)$, answering Problem 20.2 of the Kourovka Notebook affirmatively: there do exist nonabelian simple groups of Lie type that are totally 3-closed. The main result is that $\mathrm{PSL}_2(q)$ is totally 3-closed if and only if $q \geq 7$ is prime; $\mathrm{PSL}_3(q)$ is totally 3-closed if and only if $q=3$ or $q$ is prime with $q \equiv 2 \pmod 3$; $\mathrm{PSL}_4(q)$ is never totally 3-closed; and $\mathrm{PSL}_n(q)$ is not totally 3-closed for $n \geq 5$ and $\mathrm{PSL}_2(q)$0 (2608.17878). Only $\mathrm{PSL}_2(q)$1 for $\mathrm{PSL}_2(q)$2 remains open within this family.

Background and main theorem

The $\mathrm{PSL}_2(q)$3-closure $\mathrm{PSL}_2(q)$4 of a permutation group $\mathrm{PSL}_2(q)$5 is the largest subgroup of $\mathrm{PSL}_2(q)$6 with the same orbits as $\mathrm{PSL}_2(q)$7 on $\mathrm{PSL}_2(q)$8; the action is $\mathrm{PSL}_2(q)$9-closed when equality holds, and an abstract finite group is totally $q \geq 7$0-closed when every faithful finite action is $q \geq 7$1-closed. The notion goes back to Wielandt, and total closure was developed by Churikov–Praeger and Freedman–Giudici–Praeger. Prior work on the $q \geq 7$2 case showed that among nonabelian simple groups exactly six are totally 2-closed — $q \geq 7$3, $q \geq 7$4, $q \geq 7$5, $q \geq 7$6, $q \geq 7$7, and $q \geq 7$8 — all sporadic, so no simple group of Lie type is totally 2-closed. This motivated Kourovka Notebook Problem 20.2, which asks whether any Lie type simple group is totally 3-closed.

The answer is yes, in abundance: every $q \geq 7$9 with prime $\mathrm{PSL}_3(q)$0, and every $\mathrm{PSL}_3(q)$1 with prime $\mathrm{PSL}_3(q)$2, together with $\mathrm{PSL}_3(q)$3, is totally 3-closed. This is a sharp structural contrast with the 2-closed case, where the entire Lie type family fails.

The reduction framework

The proofs rest on a two-orbit criterion: since every nonsingleton orbit of a nonabelian simple group is faithful, $\mathrm{PSL}_3(q)$4 is totally 3-closed if and only if the diagonal action of $\mathrm{PSL}_3(q)$5 on $\mathrm{PSL}_3(q)$6 is 3-closed for all proper subgroups $\mathrm{PSL}_3(q)$7 (with repetition). This reduces an abstract property over all faithful actions to a finite-in-principle check on pairs of coset actions.

Three auxiliary tools handle most pairs. First, any action with base size at most two is 3-closed, and if one constituent of a pair has base size at most two and both are 3-closed, so is their union. Second, if $\mathrm{PSL}_3(q)$8, the union of two 3-closed coset actions is 3-closed. Consequently, positive results reduce to the small collection of subgroups whose coset actions lack a base of size two — chiefly parabolic-type subgroups.

The $\mathrm{PSL}_3(q)$9 classification

Proper prime powers are excluded by a Frobenius obstruction: for $q=3$0 with $q=3$1, the Frobenius map on $q=3$2 preserves the determinant square class that distinguishes the two $q=3$3-orbits on ordered triples of distinct points, so it lies in $q=3$4; when $q=3$5 is even, $q=3$6 is sharply 3-transitive, so $q=3$7. Also $q=3$8 fails since its 3-transitive degree-5 action has $q=3$9.

For primes $q$0, $q$1, Dickson's subgroup classification reduces everything to torus normalizers, exceptional subgroups $q$2, and subgroups $q$3 of the Borel. Torus normalizers and exceptional subgroups admit disjoint conjugates (the latter via a union-bound counting argument over prime-order subgroup classes), giving base size two. The Borel-fiber actions on $q$4 are shown 3-closed via a rigidity argument combining the determinant square class on projective triples with Carlitz's theorem on permutations preserving square classes of difference quotients.

The remaining primes $q$5 are handled individually using distinctive combinatorial structures: the Fano plane for $q$6, the unique 2-$q$7 biplane and the Paley tournament for $q$8, the Paley graph and scalar-fiber rigidity for $q$9, and the Perkel graph (whose full automorphism group is $q \equiv 2 \pmod 3$0, forcing the degree-57 actions to be 2-closed) for $q \equiv 2 \pmod 3$1.

The $q \equiv 2 \pmod 3$2 classification

For $q \equiv 2 \pmod 3$3, the paper uses the maximal subgroup structure ($q \equiv 2 \pmod 3$4, $q \equiv 2 \pmod 3$5, $q \equiv 2 \pmod 3$6, $q \equiv 2 \pmod 3$7), shows all coset actions outside the parabolics have base size two by counting conjugate intersections, proves the point and line actions on $q \equiv 2 \pmod 3$8 are 3-closed via the fundamental theorem of projective geometry, and handles the exceptional parabolic descendants with the paper's only computer-assisted step: a GAP program (running in under ten seconds) enumerating all 646 subgroups of the point stabilizer, isolating 24 exceptional subgroups in six conjugacy classes, and verifying $q \equiv 2 \pmod 3$9 for each. Diagonal pairs of parabolic actions are handled by a synchronization lemma: mixed triple orbits recover equality or incidence in $\mathrm{PSL}_4(q)$0, forcing the induced permutation to fix all points or all lines, hence to be trivial.

For primes $\mathrm{PSL}_4(q)$1 (so $\mathrm{PSL}_4(q)$2 and $\mathrm{PSL}_4(q)$3), the scalar-fiber actions $\mathrm{PSL}_4(q)$4 are shown 3-closed by a rigidity argument: fiber scalars $\mathrm{PSL}_4(q)$5 must satisfy $\mathrm{PSL}_4(q)$6 on noncollinear triples, forcing a global scalar $\mathrm{PSL}_4(q)$7 with $\mathrm{PSL}_4(q)$8, which is trivial since $\mathrm{PSL}_4(q)$9. Nonparabolic maximal subgroups (triangle stabilizers, Singer cycle normalizers, $\mathrm{PSL}_n(q)$0, $\mathrm{PSL}_n(q)$1) all admit disjoint conjugates via explicit constructions and counting. Parabolic descendants are handled by an intricate argument analyzing cores $\mathrm{PSL}_n(q)$2 in point stabilizers, using a Levi-separator lemma in $\mathrm{PSL}_n(q)$3 and the connectedness of the graph of ordered projective bases to propagate a global triviality.

The obstructions

Two clean obstructions give all negative results. The semilinear obstruction shows that $\mathrm{PSL}_n(q)$4 on projective space: $\mathrm{PSL}_n(q)$5 and $\mathrm{PSL}_n(q)$6 share orbits on triples (determinant classes can always be adjusted on the stabilizer of any triple), so any field automorphism, and any element of $\mathrm{PSL}_n(q)$7 when $\mathrm{PSL}_n(q)$8, lies in the 3-closure. This excludes all proper prime powers and all cases with $\mathrm{PSL}_n(q)$9 in dimension three.

The central-vector obstruction handles $n \geq 5$0, $n \geq 5$1: on the coset space $n \geq 5$2 where $n \geq 5$3, every element of $n \geq 5$4 can be matched on any ordered $n \geq 5$5-tuple with $n \geq 5$6 by an element of $n \geq 5$7, by scaling a basis vector outside the span of the tuple. Since $n \geq 5$8, the action is not $n \geq 5$9-closed for any $\mathrm{PSL}_2(q)$00. Finally, $\mathrm{PSL}_2(q)$01 fails via its action on two-subsets of $\mathrm{PSL}_2(q)$02, where every $\mathrm{PSL}_2(q)$03-orbit on triples is an $\mathrm{PSL}_2(q)$04-orbit because any odd permutation can be corrected by a transposition of two unused points.

Limitations and open questions

The paper explicitly leaves open the total 3-closedness of $\mathrm{PSL}_2(q)$05 for $\mathrm{PSL}_2(q)$06; neither obstruction applies there ($\mathrm{PSL}_2(q)$07 has no nontrivial field automorphism, $\mathrm{PSL}_2(q)$08, and the central-vector argument requires $\mathrm{PSL}_2(q)$09). Finite simple groups of other Lie types are not addressed, so the full classification of totally 3-closed Lie type groups remains open. The $\mathrm{PSL}_2(q)$10 argument depends on a GAP computation, though the authors report an independent Python implementation returning identical results, and the paper includes a report of AI assistance (the Albilich system) in generating the main argument, with subsequent independent verification and rewriting by the authors. It is also worth noting the paper's positive results rely on the classification of subgroups of $\mathrm{PSL}_2(q)$11 (Dickson, via Guralnick–Zieve) and of $\mathrm{PSL}_2(q)$12 (King), so they inherit any dependence on those classifications.

Conclusion

The paper establishes a complete classification of total 3-closedness for $\mathrm{PSL}_2(q)$13 except for $\mathrm{PSL}_2(q)$14 with $\mathrm{PSL}_2(q)$15, and in doing so provides the first affirmative answer to Kourovka Problem 20.2. The contrast with the 2-closed case — where no Lie type group qualifies — shows that the passage from $\mathrm{PSL}_2(q)$16 to $\mathrm{PSL}_2(q)$17 fundamentally changes which simple groups are totally closed, with the answer governed by arithmetic conditions on $\mathrm{PSL}_2(q)$18 relative to $\mathrm{PSL}_2(q)$19 (primality of $\mathrm{PSL}_2(q)$20, and coprimality of $\mathrm{PSL}_2(q)$21 and $\mathrm{PSL}_2(q)$22) rather than by the Lie type alone.