Maximality Degree Elements of Finite Cyclic Group Zpn, Zpmpn
Bibliographic record
Abstract
In this article, the concept of maximality degree of a finite group G, where G is cyclic group 𝑍 𝑝 𝑛 or 𝑍 𝑝 𝑚 𝑞 𝑛 is introduced and studied in details.The probability of a random subgroup of G to be maximal is measured by this quantity.For certain special kinds of finite groups, explicit formulas are obtained.We will give a value of one when the probability of 〈𝑥, 𝑦〉 ≤ 𝑚𝑎𝑥 𝐺, and a value of zero when it does not maximal sub group.This will be useful in our research to calculate the degree of probability.Several limits of degrees of maximality are also calculated.We studied three cases, the first is when 𝑝 is a prime number in 𝑍 𝑝 , the second is when 𝑝 is a prime number raised to a certain degree in 𝑍 𝑝 𝑛 , and the third case is when 𝑝 and 𝑞 are the product of two prime numbers, each of these prime numbers is raised to a certain degree in 𝑍 𝑝 𝑛 𝑞 𝑚 .We find an algorithm to compute the probability of maximality degree Pmax(G).We will use the CAP program to compute the number of maximal subgroups of group G.In this program, we will calculate the max sub groups when 𝑝, 𝑞 is a large number that is difficult to calculate manually.
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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 | 0.001 | 0.005 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.001 | 0.001 |
| Science and technology studies | 0.001 | 0.002 |
| Scholarly communication | 0.001 | 0.003 |
| Open science | 0.001 | 0.001 |
| Research integrity | 0.000 | 0.001 |
| Insufficient payload (model declined to judge) | 0.003 | 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 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".