Partial augmentations power property: A Zassenhaus Conjecture related problem
| Autor: | Leo Margolis, Ángel del Río |
|---|---|
| Přispěvatelé: | Algebra, Mathematics |
| Rok vydání: | 2019 |
| Předmět: |
Normal subgroup
Pure mathematics Finite group Algebra and Number Theory Conjecture 010102 general mathematics Zassenhaus conjecture Mathematics - Rings and Algebras 16U60 16S34 20C05 20C10 01 natural sciences Groups of units Mathematics::Group Theory Nilpotent Partial augmentation 0103 physical sciences Order (group theory) 010307 mathematical physics integral group ring 0101 mathematics Element (category theory) Mathematics - Group Theory Unit (ring theory) Mathematics - Representation Theory Group ring Mathematics |
| Zdroj: | arXiv.org e-Print Archive |
| ISSN: | 0022-4049 |
| DOI: | 10.1016/j.jpaa.2018.12.018 |
| Popis: | Zassenhaus conjectured that any unit of finite order in the integral group ring $\mathbb{Z}G$ of a finite group $G$ is conjugate in the rational group algebra of $G$ to an element in $\pm G$. We review the known weaker versions of this conjecture and introduce a new condition, on the partial augmentations of the powers of a unit of finite order in $\mathbb{Z}G$, which is weaker than the Zassenhaus Conjecture but stronger than its other weaker versions. We prove that this condition is satisfied for units mapping to the identity modulo a nilpotent normal subgroup of $G$. Moreover, we show that if the condition holds then the HeLP Method adopts a more friendly form and use this to prove the Zassenhaus Conjecture for a special class of groups. Comment: 14 pages. A gap fixed and some typos corrected |
| Databáze: | OpenAIRE |
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