关于（弱）小群的性质
On properties of (weakly) small groups
摘要：
一个群是小的，如果它对每个自然数n在空集上有可数多个完全$n$ -型。更一般地说，群$G$是弱小的，如果它在G的每个有限子集上有可数个完全1-型。本文证明了在弱子群中，参数在一个弱生成的代数闭包中的子群，其迹在任意一个弱生成的代数闭包中满足降链条件.一个无限弱子群有一个无限阿贝尔子群，它可能是不可定义的。小幂零群群是可定义可除群与可定义有界指数群的中心积。在一个理论简单的群中，任何两两交换元素的集合都包含在一个可定义的有限交换子群中。第一个推论：一个理论简单的弱小群有一个无限可定义的有限交换子群。其次，在一个理论简单的群中，一个导长为n的正规可解群A包含在一个类n的A-可定义几乎可解群中。
英文摘要：
A group is small if it has countably many complete $n$-types over the empty set for each natural number n. More generally, a group $G$ is weakly small if it has countably many complete 1-types over every finite subset of G. We show here that in a weakly small group, subgroups which are definable with parameters lying in a finitely generated algebraic closure satisfy the descending chain conditions for their traces in any finitely generated algebraic closure. An infinite weakly small group has an infinite abelian subgroup, which may not be definable. A small nilpotent group is the central product of a definable divisible group with a definable one of bounded exponent. In a group with simple theory, any set of pairwise commuting elements is contained in a definable finite-by-abelian subgroup. First corollary : a weakly small group with simple theory has an infinite definable finite-by-abelian subgoup. Secondly, in a group with simple theory, a normal solvable group A of derived length n is contained in an A-definable almost solvable group of class n.
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