Enumerating Groups Acting Regularly on a <i>d</i> -Dimensional Cube
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Bibliographic record
Abstract
In the survey article [2 Li , C. H. ( 2002 ). On isomorphisms of finite Cayley graphs – a survey . Discrete Mathematics 256 ( 1–2 ): 301 – 334 .[Crossref], [Web of Science ®] , [Google Scholar]] it was noted, among many other open problems, that the classification of the groups acting regularly on a d-dimensional cube Γ is unsettled. In other words, the classification of the finite groups G such that Cay(G, S) ≅ Γ, for some subset S of G, is still unknown. In this article, we prove that there are at least 2 d 2/64 − (d/2)log2(d/2) nonisomorphic 2-groups of Frattini class 2 acting regularly on a d-dimensional cube. Other relevant results are presented. As a corollary of our result, we remark that the symmetric group Sym(n) on n symbols contains at least 2 n 2/256 − (n/4)log2(n/4) subgroups up to isomorphism. In particular, we recall that in [4 Pyber , L. ( 1993 ). Enumeranting finite groups of given order . Ann. Math. 137 ( 2 ): 203 – 220 .[Crossref], [Web of Science ®] , [Google Scholar]] it was proved that the total number of subgroups of Sym(n) is at most 2 cn 2 , for c = log224.
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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 | 0.002 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.001 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.001 | 0.000 |
| Research integrity | 0.000 | 0.001 |
| 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 it