标题： 群环中的单位与克莱因四元或二面体缺陷块 显示英文标题

标题： Units in group rings and blocks of Klein four or dihedral defect

摘要： 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 $\mathbb{Z}_2G$. 这改进了“格方法”，该方法考虑模素数 $p$ 的约简，但对于 $p=2$ 的使用有限，这主要是由于 $1\equiv -1 \ (\textrm{mod }2)$ 的事实。 我们的方法在 $\mathbb Z_2 G$ 具有缺陷群为克莱因四元群或阶数为 $8$ 的二面体群的块的情况下得出结果。 这使我们能够反驳具有陪衬 $\operatorname{PSL}(2,p^f)$ 的几乎简单群中阶为 $2p$ 的单位的存在性，其中 $p^f\equiv \pm 3 \ (\textrm{mod } 8)$，并肯定地回答了许多此类群的素数图问题。 显示英文摘要

摘要： 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.