Bibliographic record
Abstract
We exhibit presentations of the Von Dyck groups $D(2, 3, m), \ m\ge 3$, in terms of two generators of order $m$ satisfying three relations, one of which is Artin's braid relation. By dropping the relation which fixes the order of the generators we obtain the universal covering groups of the corresponding Von Dyck groups. In the cases $m=3, 4, 5$, these are respectively the double covers of the finite rotational tetrahedral, octahedral and icosahedral groups. When $m\ge 6$ we obtain infinite covers of the corresponding infinite Von Dyck groups. The interesting cases arise for $m\ge 7$ when these groups act as discrete groups of isometries of the hyperbolic plane. Imposing a suitable third relation we obtain three-relator presentations of $\text{PSL}(2,m)$. We discover two general formulas presenting these as factors of $D(2, 3, m)$. The first one works for any odd $m$ and is essentially equivalent to the shortest known presentation of Sunday [J. Sunday, Presentations of the groups ${\rm SL}(2,\,m)$ and ${\rm PSL}(2,\,m)$, Canadian J. Math., 24 (1972) 1129--1131]. The second applies to the cases $m\equiv\pm 2\ (\text{mod}\ 3)$, $m\equiv \hskip -9pt/ \ 11(\text{mod}\ 30)$, and is substantively shorter. Additionally, by random search, we find many efficient presentations of finite simple Chevalley groups PSL($2,q$) as factors of $D(2, 3, m)$ where $m$ divides the order of the group. The only other simple group that we found in this way is the sporadic Janko group $J_2$.
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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 | 0.003 | 0.002 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.000 |
| Bibliometrics | 0.001 | 0.002 |
| Science and technology studies | 0.001 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.005 | 0.003 |
| Research integrity | 0.000 | 0.001 |
| Insufficient payload (model declined to judge) | 0.025 | 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".