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

近三十年的研究结果表明，有限呈现群的子群结构非常复杂。例如，已知有限生成的元数据组是有限呈现的元数据组的子组。为了进一步研究有限表示群的复杂性，确定具有特定属性的有限生成元群是否可以嵌入具有相同属性的有限表示元群是很有意义的。研究者最初的目的是证明一个一般的陈述，即如果G是一个有限生成的剩余幂零（或剩余无扭幂零）代谢群，那么G是一个有限呈现的也是剩余幂零（或剩余无扭幂零）的代谢群的子群。这似乎是一个复杂的问题，就像解决自由的metabelian情况所需的机制所证明的那样。在自由亚元群的特殊情况下，已知有限生成的自由亚元群是残零幂和残零无扭幂零的。研究者已经证明了一个有限生成的自由代谢群可以嵌入到一个有限呈现的代谢群中，这个代谢群也是残零的。此外，一个有限生成的自由亚元群可以嵌入到一个有限呈现的也是剩余无扭转的幂零亚元群中。为了得到这些结果，研究者开发了一种工具，它是由矩阵表示得到的有限表示的稳定群的残差零幂和残差无扭转零幂的充分条件。矩阵项属于分数的指定组和指定模块。为了证明能满足充分条件，利用了理想的S-隔离子的概念，其中S是一个乘法半群。调查人员正试图开发进一步的见解和额外的工具，试图解决上面提到的最初问题。一个需要考虑的扩展是让矩阵条目属于某个模块的因子群，并确定所得到的矩阵群是否剩余无扭转幂零。

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.