The intersection density of non-quasiprimitive groups of degree 3p
Bibliographic record
Abstract
The intersection density of a finite transitive group G ≤ Sym ( Ω ) is the rational number ρ ( G ) given by the ratio between the maximum size of a subset of G in which any two permutations agree on some elements of Ω and the order of a point stabilizer of G . In 2022, Meagher asked whether ρ ( G ) ∈ { 1 , 3 2 , 3 } for any transitive group G of degree 3 p , where p ≥ 5 is an odd prime. If G ≤ Sym ( Ω ) is transitive such that | Ω | = 3 p , then it is known that ρ ( G ) = 1 whenever (a) G is primitive or (b) G is imprimitive and admits a block of size p or at least two G -invariant partitions of Ω. In order to answer Meagher's question, it is left to analyze the intersection density of groups G admitting a unique G -invariant partition B whose blocks are of size 3. If G is such a group and G ‾ is the group induced by the action of G on B , then we denote the kernel of the canonical epimorphism G → G ‾ by ker ( G → G ‾ ) . The subgroup ker ( G → G ‾ ) is trivial if and only if G is quasiprimitive. It is shown in this paper that the answer to Meagher's question is affirmative for non-quasiprimitive groups of degree 3 p , unless possibly when p = q + 1 is a Fermat prime and Ω admits a unique G -invariant partition B whose blocks are of size 3 such that the induced action G ‾ is an almost simple group with socle equal to PSL 2 ( q ) .
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How this classification was reachedexpand
Full frame distilled prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. Learned from the 10,348 direct Codex labels and 10,348 direct Gemma labels. Candidate is the union of thresholded teacher heads; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels or direct frontier model labels.
Codex and Gemma teacher scores by category
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.002 | 0.001 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.000 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.000 | 0.000 |
Machine scores (provisional)
The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.
Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from itClassification
machine, unvalidatedMachine predicted; a candidate call from one teacher head, not a consensus.
How this classification was reached, model by model and score by score, is at the end of the page under "How this classification was reached".