THE NEIGHBOURHOOD OF DIHEDRAL 2-GROUPS
Bibliographic record
Abstract
Abstract. We examine two particular constructions that derive from a 2-group G = G(·) another 2-group G(∗) for the case when G(·) is one of D2n, SD2n, Q2n. The constructions (the cyclic one and the dihedral one) have the property that x ∗ y = x · y for exactly 3/4 of all pairs (x, y) ∈ G × G. If G(◦) and G(∗) are such 2-groups that x ◦ y � = x ∗ y for less then a quarter of all pairs (x, y) ∈ G × G, then G(◦) and G(∗) are isomorphic [4]. This makes of a special interest situations when G(◦) and G(∗) are not isomorphic and x ◦ y � = x ∗ y for exactly one quarter of all pairs. Say that groups G1 and G2 can be placed in quarter distance, if there exist G(◦) ∼ = G1 and G(∗) ∼ = G2 of the above property. It seems to be a difficult task to decide for which G1 and G2 such G(◦) and G(∗) exist. Nevertheless, there exist general constructions that allow to derive from a group G = G(·) another group G(∗) such that x · y = x ∗ y for exactly three quarters of all pairs (x, y) ∈ G × G. The constructions we shall examine here are called cyclic and dihedral. They are described in [5] and it seems that all presently known pairs of 2-groups that can be placed in quarter distance can be interpreted in terms of these constructions.
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How this classification was reachedexpand
Full frame machine prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. The Gemma side is a direct model label for every work in the frame, read from the title-only record. The Codex side is a classifier learned from the 10,348 direct Codex labels and calibrated to design-weighted sample rates; fields without enough sample support carry no Codex call. Candidate is the union of the two sides; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels.
Distilled classifier scores by category (both heads)
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.000 | 0.001 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.001 | 0.000 |
| Science and technology studies | 0.001 | 0.001 |
| Scholarly communication | 0.001 | 0.002 |
| Open science | 0.001 | 0.002 |
| Research integrity | 0.001 | 0.001 |
| Insufficient payload (model declined to judge) | 0.007 | 0.001 |
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 source (direct Gemma or distilled Codex), 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".