小维解析pro-p群
Analytic pro-p groups of small dimensions
摘要：
根据Lazard，每个p-adic李群包含一个开的pro-p子群，它是饱和的。这可以被看作是p-adic李理论的起点，因为人们可以自然地将每个可饱和pro-p群G与p-adic整数上的李格L（G）联系起来。饱和pro-p群的基本特征包括它们是挠自由的和p-adic解析的。本文证明了在小维数下的一个逆结果：维数小于p的无挠p-adic解析pro-p群是饱和的.这导致了有用的结果和有趣的问题。例如，我们给出了p > 3的3维可解无挠p-adic解析pro-p群的有效分类。我们的方法通过李理论是可比的使用拉扎德的对应分类有限p-群的小订单。
英文摘要：
According to Lazard, every p-adic Lie group contains an open pro-p subgroup which is saturable. This can be regarded as the starting point of p-adic Lie theory, as one can naturally associate to every saturable pro-p group G a Lie lattice L(G) over the p-adic integers. Essential features of saturable pro-p groups include that they are torsion-free and p-adic analytic. In the present paper we prove a converse result in small dimensions: every torsion-free p-adic analytic pro-p group of dimension less than p is saturable. This leads to useful consequences and interesting questions. For instance, we give an effective classification of 3-dimensional soluble torsion-free p-adic analytic pro-p groups for p > 3. Our approach via Lie theory is comparable with the use of Lazard's correspondence in the classification of finite p-groups of small order.
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