论文标题
游离多努力组和Magnus属性
Free polynilpotent groups and the Magnus property
论文作者
论文摘要
一个人是由自由小组的经典结果激励的，一个人说，$ g $具有以下情况，如果有以下内容：每当两个元素产生相同的$ g $的正常亚组时，它们都是共轭或$ g $中的偶联。 发现哪些相对自由的组显示Magnus属性是一个自然的问题。我们证明，任何给定类行的免费polynilpotent组具有Magnus属性，并且只有当时最多是$ 2 $的nilpotent。为此，我们更普遍地探索了Magnus属性，并在可溶性群体中探索了新技术，包括建立和反驳该属性。我们还证明，当且只有当时最多只有2美元的$ 2 $的情况下，只有当时它是Magnus的属性，我们还证明了一个免费的中心（给定类行的polynilpotent）。 在途中，我们显示了与Magnus属性的任何处方的Nilpotency类的$ 2 $生成的Nilpotent组（具有非平凡的扭转）。类似的例子是有限生成的无扭转的无扭矩nilpotent组，但我们构建了一个$ 4 $的无扭转，无扭转，$ 3 $ 3 $ nilpotent hirsch Length $ 9 $的nilpotent Group $ 9 $。此外，使用Magnus特性的弱变体和超强构建，我们确定了具有Magnus属性的任何处方nilpotency类的Metabelian，无扭转，无扭力的nilpotent组。
Motivated by a classic result for free groups, one says that a group $G$ has the Magnus property if the following holds: whenever two elements generate the same normal subgroup of $G$, they are conjugate or inverse-conjugate in $G$. It is a natural problem to find out which relatively free groups display the Magnus property. We prove that a free polynilpotent group of any given class row has the Magnus property if and only if it is nilpotent of class at most $2$. For this purpose we explore the Magnus property more generally in soluble groups, and we produce new techniques, both for establishing and for disproving the property. We also prove that a free centre-by-(polynilpotent of given class row) group has the Magnus property if and only if it is nilpotent of class at most $2$. On the way, we display $2$-generated nilpotent groups (with non-trivial torsion) of any prescribed nilpotency class with the Magnus property. Similar examples of finitely generated, torsion-free nilpotent groups are hard to come by, but we construct a $4$-generated, torsion-free, class-$3$ nilpotent group of Hirsch length $9$ with the Magnus property. Furthermore, using a weak variant of the Magnus property and an ultraproduct construction, we establish the existence of metabelian, torsion-free, nilpotent groups of any prescribed nilpotency class with the Magnus property.
