Let G be a connected Lie group. An unrefinable chain of G is defined to be a chain of subgroups $$G = G_0> G_1> \cdots > G_t = 1$$ G = G 0 > G 1 > ⋯ > G t = 1 , where each $$G_i$$ G i is a maximal connected subgroup of $$G_{i-1}$$ G i - 1 . In this paper, we introduce the notion of the length (respectively, depth) of G , defined as the maximal (respectively, minimal) length of such a chain, and we establish several new results for compact groups. In particular, we compute the exact length and depth of every compact simple Lie group, and draw conclusions for arbitrary connected compact Lie groups G . We obtain best possible bounds on the length of G in terms of its dimension, and characterize the connected compact Lie groups that have equal length and depth. The latter result generalizes a well known theorem of Iwasawa for finite groups. More generally, we establish a best possible upper bound on $$\dim G'$$ dim G ′ in terms of the chain difference of G , which is its length minus its depth.

中文翻译：

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紧李群的长度和深度

设 G 为连通李群。G 的不可精炼链定义为子群链 $$G = G_0> G_1> \cdots > G_t = 1$$ G = G 0 > G 1 > ⋯ > G t = 1 ，其中每个 $$G_i$ $G i 是 $$G_{i-1}$$ G i - 1 的最大连通子群。在本文中，我们引入了 G 的长度（分别为深度）的概念，定义为这样一个链的最大（分别为，最小）长度，并且我们为紧群建立了几个新的结果。特别地，我们计算每个紧致简单李群的精确长度和深度，并得出任意连通紧致李群 G 的结论。我们根据维度获得 G 的长度的最佳可能界限，并表征具有相等长度和深度的相连紧李群。后一个结果推广了一个著名的 Iwasawa 定理，用于有限群。更一般地，我们根据 G 的链差（即其长度减去其深度）建立 $$\dim G'$$ dim G ' 的最佳可能上限。