Divisible rigid groups
Divisible rigid groups
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可分刚性群
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发表时间：
2008
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通讯作者：
N. Romanovskii
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作者：
N. Romanovskii
一个可解群G是刚性的，如果它包含形式为G=G1>；G2>；…的正规级数>；gp>；gp+1=1，其商Gi/Gi+1是交换的，且作为右ℤ[G/Gi]-模是挠自由的。刚性群的概念出现在研究接近自由可解群上的代数几何时。在所有刚性群的类中，我们区分其商GI/GI+1的元素可被各自的群环Z[G/GI]的任何元素整除的可除群。可以合理地假设可分刚性群上的代数几何是相当好的结构。研究了这类群的抽象性质。证明了在每一个以G为子群的可分刚性群H中，都存在一个包含G的极小可分子群，我们称之为G在H中的可除闭包。证明了直到G-同构，才能唯一地定义一个可除完备。
A soluble group G is rigid if it contains a normal series of the form G = G1 > G2 > … > Gp > Gp+1 = 1, whose quotients Gi/Gi+1 are Abelian and are torsion-free as right ℤ[G/Gi]-modules. The concept of a rigid group appeared in studying algebraic geometry over groups that are close to free soluble. In the class of all rigid groups, we distinguish divisible groups the elements of whose quotients Gi/Gi+1 are divisible by any elements of respective groups rings Z[G/Gi]. It is reasonable to suppose that algebraic geometry over divisible rigid groups is rather well structured. Abstract properties of such groups are investigated. It is proved that in every divisible rigid group H that contains G as a subgroup, there is a minimal divisible subgroup including G, which we call a divisible closure of G in H. Among divisible closures of G are divisible completions of G that are distinguished by some natural condition. It is shown that a divisible completion is defined uniquely up to G-isomorphism.