Artinian groups of large cardinality

Samuel M. Corson, Saharon Shelah
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Abstract

A group is Artinian if there is no infinite strictly descending chain of subgroups. Ol'shanskii has asked whether there are Artinian groups of arbitrarily large cardinality. We reduce this problem to an analogous question, regarding universal algebras, asked by J\'onsson in the 1960s. We provide Artinian groups of cardinality $\aleph_n$ for each natural number $n$. We also give a consistent strong negative answer to the question of Ol'shanskii (from a large cardinal assumption) as well as a consistent positive answer. Thus, answers to the questions of Ol'shanskii and J\'onsson are independent of set theory.
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大卡数阿蒂尼群
如果不存在无限严格降序的子群链,那么这个群就是阿蒂尼亚群。奥尔尚斯基曾提出是否存在任意大心数的阿蒂尼亚群。我们把这个问题还原为一个类似的问题,即 J\'onsson 在 20 世纪 60 年代提出的关于普遍代数的问题。我们为每个自然数 $n$ 提供了心数为 $\aleph_n$ 的阿蒂尼亚群。我们还对奥尔尚斯基的问题给出了一致的强否定答案(来自大心性假设)和一致的肯定答案。因此,对奥尔尚斯基和约翰逊问题的回答是独立于设定理论的。
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