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 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 | 0.001 | 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) | 0.001 | 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".