Mathematical Sciences: Subgroups of Residually Torsion- Free Nilpotent Metabelian Groups

近三十年来的研究结果表明，群的亚群结构非常复杂。 例如，已知的是，一个非生成的亚阿贝尔群是一个非表示的亚阿贝尔群的子群。 为了更进一步地研究群的复杂性，确定一个具有特定性质的群生成的亚阿贝尔群是否可以嵌入到具有相同性质的群中是有意义的。 研究者的最初目标是证明一般陈述，即如果G是一个生成的剩余幂零（或剩余无挠幂零）亚阿贝尔群，则G是一个表示的亚阿贝尔群的子群，它也是剩余幂零（或剩余无挠幂零）的。 这似乎是一个复杂的问题，正如自由亚阿贝尔情况下解决它所需的机器所证明的那样。 在自由亚阿贝尔群的特殊情况下，已知n-生成的自由亚阿贝尔群是剩余幂零和剩余无挠幂零的。 研究者已经证明了一个可生成的自由亚阿贝尔群可以嵌入到一个可表示的亚阿贝尔群中，该亚阿贝尔群也是剩余幂零的。 此外，一个无挠生成的自由亚阿贝尔群可以嵌入到一个无挠表示的亚阿贝尔群中，该亚阿贝尔群也是剩余无挠幂零的。 由调查员开发的工具之一，以获得这些结果是一个充分条件的剩余零和剩余无挠零的某些random提出的亚阿贝尔群所产生的矩阵表示的亚阿贝尔群。 矩阵元素属于指定的组和指定的分数模。 为了证明这个充分条件可以被满足，我们利用了理想的S-隔离子的概念，这里S是一个乘法半群。 该委员会正在试图开发进一步的见解和其他工具，以解决上述最初的问题。 要考虑的一个扩展是让矩阵元素属于某些因子组的模块，并确定是否产生矩阵组是剩余无挠幂零。

Research results over the last thirty years have indicated that finitely presented groups are very complex in their subgroup structure. For instance, it is known that a finitely generated metabelian group is a subgroup of a finitely presented metabelian group. In order to examine still further the complexity of finitely presented groups, it is of interest to determine whether a finitely generated metabelian group with a specific property can be embedded in a finitely presented metabelian group with the same property. The initial objective of the investigator had been to prove the general statement that if G is a finitely generated residually nilpotent (or residually torsion-free nilpotent) metabelian group, then G is a subgroup of a finitely presented metabelian group that is also residually nilpotent (or residually torsion-free nilpotent). This appears to be a complicated problem, as evidenced by the machinery required to solve it for the free metabelian case. In the special case of free metabelian groups it is known that finitely generated free metabelian groups are residually nilpotent and residually torsion-free nilpotent. The investigator has proved that a finitely generated free metabelian group can be embedded in a finitely presented metabelian group that is also residually nilpotent. Furthermore, a finitely generated free metabelian group can be embedded in a finitely presented metabelian group that is also residually torsion-free nilpotent. One of the tools developed by the investigator to obtain these results is a sufficient condition for the residual nilpotence and residual torsion-free nilpotence of certain finitely presented metabelian groups that arise from a matrix representation for metabelian groups. The matrix entries belong to specified groups and specified modules of fractions. To show the sufficient condition can be satisfied, use is made of the notion of S-isolator of an ideal, where S is a multiplicative semigroup. The investigat or is attempting to develop further insight and additional tools with which to try to solve the initial problem mentioned above. An extension to be considered is to let the matrix entries belong to certain factor groups of modules and to determine whether the resulting matrix group is residually torsion-free nilpotent.