A Subnormal Closure Version and Proof of the Guralnick-Tracey Theorem on Proper Normal Subgroup Containment
Bibliographic record
Abstract
In [8] we compared Flavell [6] and Guralnick-Tracey [7] criteria for a test subgroup K of a finite group G to be contained in a proper normal subgroup of G. In order to find useful examples one must first consider non-nilpotent groups with some but relatively few normal subgroups and with several layers in their lattice of subgroups. This precluded an effective approach by hand calculation, so instead we found suitable groups by considering the Shephard-Todd finite unitary reflection groups of rank 2 contained in U(2,C), and other related large groups in GL(4,C), which we represented using the Maplesoft™ built-in interactive matrix algebra over the complex numbers and generators from [2] and [3]. In these computational investigations we showed that the Guralnick-Tracey criteria were strictly stronger than the Flavell criteria, in that they were much more successful at detecting containment of a test subgroup in a proper normal subgroup in the many examples we considered. Furthermore, we verified computationally in all these examples that the descending normal closure of the test subgroup used by Guralnick-Tracey was identical to its subnormal closure. This lead to the observation that the Guralnick-Tracey theorem can be restated in terms of the subnormal closure of the test subgroup, which provides a new more transparent unification of the roles of the intimately related theorems of Flavell [6] and Wielandt [5]. In this paper we give a subnormal closure version of the Guralnick-Tracey theorem and present a new computationally guided proof while simultaneously displaying our computational examples corresponding to the various cases.
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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 | 0.002 | 0.009 |
| Meta-epidemiology (narrow) | 0.001 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.002 | 0.001 |
| Science and technology studies | 0.002 | 0.006 |
| Scholarly communication | 0.002 | 0.005 |
| Open science | 0.002 | 0.004 |
| Research integrity | 0.001 | 0.004 |
| Insufficient payload (model declined to judge) | 0.008 | 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".