Universal theories for rigid soluble groups
Universal theories for rigid soluble groups
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刚性可溶基团的通用理论
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发表时间：
2012
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通讯作者：
N. Romanovskii
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作者：
A. Myasnikov;N. Romanovskii
一个群被称为 p-刚性群，其中 p 是自然数，如果它具有形式为 G = G1 > G2 > … > Gp > Gp+1 = 1 的正规级数，其商 Gi/Gi+1 是阿贝尔分布，并且当被视为 $ mathbb{Z} 时是无挠的$[G/Gi]-模块。刚性基团的例子是游离可溶基团。我们指出了一个普遍公理的递归系统，区分 p-可溶群中的 p-刚性群。证明了如果F是自由p可溶群，G是任意p-刚性群，W是p个无限循环群的迭代花圈积，则这些群的∀-理论满足包含$ mathcal{A}(F)supseteq mathcal{A}(G)supseteq mathcal{A}(W)$。我们构造一个 ∃-公理来区分普遍等价于 W 的 p-刚性群。任意 p-刚性群嵌入到可分分解的 p-刚性群 M = M(α1,…,αp) 中。后一组因式分解为阿贝尔群 A1A2…Ap 的半直积，在这种情况下，其刚性级数的每个商 Mi/Mi+1 与 Ai 同构，并且是环 $ mathbb{Z} $[M/Mi] 上的阶 αi 的可整除模。我们指定一个递归公理系统，区分 M 群中与 M 普遍等价的那些。因此，指出 M 的普遍理论以及 M 中的常数是可判定的。相比之下，具有常数的 W 的普遍理论是不可判定的。
A group is said to be p-rigid, where p is a natural number, if it has a normal series of the form G = G1 > G2 > … > Gp > Gp+1 = 1, whose quotients Gi/Gi+1 are Abelian and are torsion free when treated as $ mathbb{Z} $[G/Gi]-modules. Examples of rigid groups are free soluble groups. We point out a recursive system of universal axioms distinguishing p-rigid groups in the class of p-soluble groups. It is proved that if F is a free p-soluble group, G is an arbitrary p-rigid group, and W is an iterated wreath product of p infinite cyclic groups, then ∀-theories for these groups satisfy the inclusions $ mathcal{A}(F) supseteq mathcal{A}(G) supseteq mathcal{A}(W) $. We construct an ∃-axiom distinguishing among p-rigid groups those that are universally equivalent to W. An arbitrary p-rigid group embeds in a divisible decomposed p-rigid group M = M(α1,…, αp). The latter group factors into a semidirect product of Abelian groups A1A2…Ap, in which case every quotient Mi/Mi+1 of its rigid series is isomorphic to Ai and is a divisible module of rank αi over a ring $ mathbb{Z} $[M/Mi]. We specify a recursive system of axioms distinguishing among M-groups those that are Muniversally equivalent to M. As a consequence, it is stated that the universal theory of M with constants in M is decidable. By contrast, the universal theory of W with constants is undecidable.