论文标题
整体圈环的单位组的Abelianization
Abelianization of the unit group of an integral group ring
论文作者
论文摘要
对于有限的组$ g $和$ u：= u（\ mathbb {z} g）$，$ g $的整体组环的组组，我们研究了$ g $的结构对$ u $的$ u/u'$'$ $ u $的含义。我们提出有关$ g/g'$的指数与$ u/u'$指数之间的连接的问题，以及$ z（u）$的无扭矩零件的等级，$ u $的中心和$ u/u'$。我们表明，源自$ \ mathbb {z} g $单位的已知通用构造的单元在投影率从$ u $到$ u/u'$的投影中得到很好的行为，我们的问题对许多示例都有积极的答案。然后，我们展示了一个明确的例子，该例子表明，无扭转部件的一般性语句不存在，这也回答了[bjj $^+$ 18]中的问题。
For a finite group $G$ and $U: = U(\mathbb{Z}G)$, the group of units of the integral group ring of $G$, we study the implications of the structure of $G$ on the abelianization $U/U'$ of $U$. We pose questions on the connections between the exponent of $G/G'$ and the exponent of $U/U'$ as well as between the ranks of the torsion-free parts of $Z(U)$, the center of $U$, and $U/U'$. We show that the units originating from known generic constructions of units in $\mathbb{Z}G$ are well-behaved under the projection from $U$ to $U/U'$ and that our questions have a positive answer for many examples. We then exhibit an explicit example which shows that the general statement on the torsion-free part does not hold, which also answers questions from [BJJ$^+$18].
