A Subnormal Closure Version and Proof of the Guralnick-Tracey Theorem on Proper Normal Subgroup Containment
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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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| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.001 | 0.000 |
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| Bibliometrics | 0.000 | 0.001 |
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| Open science | 0.000 | 0.000 |
| Research integrity | 0.000 | 0.001 |
| Insufficient payload (model declined to judge) | 0.002 | 0.000 |
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