Units in group rings and blocks of Klein four or dihedral defect
| Autor: | Eisele, Florian, Margolis, Leo |
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| Rok vydání: | 2024 |
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| Druh dokumentu: | Working Paper |
| Popis: | We obtain restrictions on units of even order in the integral group ring $\mathbb{Z}G$ of a finite group $G$ by studying their actions on the reductions modulo $4$ of lattices over the $2$-adic group ring $\mathbb{Z}_2G$. This improves the "lattice method" which considers reductions modulo primes $p$, but is of limited use for $p=2$ essentially due to the fact that $1\equiv -1 \ (\textrm{mod }2)$. Our methods yield results in cases where $\mathbb Z_2 G$ has blocks whose defect groups are Klein four groups or dihedral groups of order $8$. This allows us to disprove the existence of units of order $2p$ for almost simple groups with socle $\operatorname{PSL}(2,p^f)$ where $p^f\equiv \pm 3 \ (\textrm{mod } 8)$ and to answer the Prime Graph Question affirmatively for many such groups. Comment: 17 pages, comments welcome |
| Databáze: | arXiv |
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