Coproducts of rigid groups
Coproducts of rigid groups
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刚性群的联积
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发表时间：
2011
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通讯作者：
N. Romanovskii
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作者：
N. Romanovskii
令 ε = (ε1, ..., εm) 为由 0 和 1 组成的元组。假设群 G 具有形式为 G = G1 ≥ G2 ≥ 的正规级数。 。 。 ≥ Gm ≥ Gm+1 = 1，其中 εi = 1 时 Gi > Gi+1，εi = 0 时 Gi = Gi+1，并且该级数的所有因子 Gi/Gi+1 都是阿贝尔因子，并且作为右 ℤ[G/Gi]-模是无扭的。这样的级数（如果存在）由群 G 和元组 ε 唯一定义。我们将指定系列的 G 称为分级为 ε 的刚性 m 分级群。在派生长度为 m 的自由可解群中，一系列派生子群满足上述条件。我们定义刚性 m 分级群态射的概念。证明了刚性 m 分级群的范畴包含余积，并且我们展示了如何构造两个给定刚性 m 分级群的余积 G°H。还指出，如果 G 是分级为 (1, 1, ..., 1) 的刚性 m 分级群，且 F 是导出长度为 m 的自由可解群，基为 {x1, ..., 1, 1, ..., 1)。 。 。 , xn}，则 G°F 是变量 x1, 中的仿射空间 Gn 的坐标群。 。 。 , xn 并且这个空间在 Zariski 拓扑中是不可约的。
Let ε = (ε1, . . . , εm) be a tuple consisting of zeros and ones. Suppose that a group G has a normal series of the form G = G1 ≥ G2 ≥ . . . ≥ Gm ≥ Gm+1 = 1, in which Gi > Gi+1 for εi = 1, Gi = Gi+1 for εi = 0, and all factors Gi/Gi+1 of the series are Abelian and are torsion free as right ℤ[G/Gi]-modules. Such a series, if it exists, is defined by the group G and by the tuple ε uniquely. We call G with the specified series a rigid m-graded group with grading ε. In a free solvable group of derived length m, the above-formulated condition is satisfied by a series of derived subgroups. We define the concept of a morphism of rigid m-graded groups. It is proved that the category of rigid m-graded groups contains coproducts, and we show how to construct a coproduct G◦H of two given rigid m-graded groups. Also it is stated that if G is a rigid m-graded group with grading (1, 1, . . . , 1), and F is a free solvable group of derived length m with basis {x1, . . . , xn}, then G◦F is the coordinate group of an affine space Gn in variables x1, . . . , xn and this space is irreducible in the Zariski topology.