To a compact Lie group G one can associate a space E(2;G) akin to the poset of cosets of abelian subgroups of a discrete group. The space E(2;G) was introduced by Adem, F. Cohen and Torres-Giese, and subsequently studied by Adem and Gómez, and other authors. In this short note, we prove that G is abelian if and only if π_i(E(2;G)) = 0 for i = 1; 2; 4. This is a Lie group analogue of the fact that the poset of cosets of abelian subgroups of a discrete group is simply connected if and only if the group is abelian.

中文翻译：

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李群中阿贝尔子群的高生成

一个紧李群G可以关联一个空间E (2; G ) 类似于离散群的阿贝尔子群的陪集的偏序集。空间E (2; G ) 由 Adem、F. Cohen 和 Torres-Giese 引入，随后由 Adem 和 Gómez 以及其他作者研究。在这个简短的注释中，我们证明 G 是阿贝尔的当且仅当π _i ( E (2; G )) = 0 for i = 1; 2; 4. 这是一个李群类比，当且仅当该群是阿贝尔群时，离散群的阿贝尔子群的陪集的偏序集是简单连通的。