Abstract

We use bicyclic units to give an explicit construction of a subgroup of $\mathcal{U}\mathbb{Z}G$ isomorphic to the free product of two free abelian groups of rank two, assuming that G is a finite nilpotent group and it contains an element g of odd prime order such that the subgroup $〈g〉$ is not normal in G. To do this we first construct a subgroup isomorphic to the desired free product inside $\text{GL}(2,\mathbb{C})$ and then we find a nontrivial matrix representation of a subgroup of $\mathcal{U}\mathbb{Z}G$ generated by some bicyclic units and their conjugations under the involution of $\mathbb{Z}G.$ We show that for an arbitrary finite group G our construction need not lead to a free product. At the end we shortly discuss possibility of constructing subgroups isomorphic to the free product of two free abelian groups of rank p − 1 for p > 3 in a similar way.

摘要

我们使用双环单元给出一个子群的显式构造 $\mathcal{你}\mathbb{Z}G$同构于两个二阶自由阿贝尔群的自由积，假设G是一个有限幂零群，并且它包含一个奇素数阶的元素g，使得子群$〈G〉$在G 中不正常。为此，我们首先在内部构造一个与所需自由积同构的子群$\text{GL}(2,\mathbb{C})$ 然后我们找到一个子群的非平凡矩阵表示 $\mathcal{你}\mathbb{Z}G$ 由一些双环单元及其对合下的共轭产生 $\mathbb{Z}G.$我们表明，对于任意有限群G，我们的构造不需要导致自由乘积。最后，我们简短地讨论 了以类似的方式构造与p > 3的秩为p  − 1的两个自由阿贝尔群的自由积同构的子群的可能性。