For an arbitrary real connected Lie group G we define $$\mathrm {p}(G)$$ p ( G ) as the maximal integer p such that $$\mathbb {Z}^p$$ Z p is isomorphic to a discrete subgroup of G and $$\mathrm {q}(G)$$ q ( G ) is the maximal integer q such that $$\mathbb {R}^q$$ R q is isomorphic to a closed subgroup of G . The aim of this paper is to investigate properties of these two invariants. We will show that if G is a noncompact connected Lie group, then $$1\le \mathrm {q}(G)\le \mathrm {p}(G)\le \dim (G/K)$$ 1 ≤ q ( G ) ≤ p ( G ) ≤ dim ( G / K ) where K is a maximal compact subgroup of G . In the cases when G is an exponential Lie group or G is a connected nilpotent Lie group, we give explicit relationships between these two invariants and a well known Lie algebra invariant $$\mathcal M(\mathfrak {g})$$ M ( g ) , i.e. the maximum among the dimensions of abelian subalgebras of the Lie algebra $$\mathfrak {g}:=\mathrm {Lie}(G)$$ g : = Lie ( G ) .

中文翻译：

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一类连通李群的极大阿贝尔无扭子群

对于任意实连通李群 G，我们定义 $$\mathrm {p}(G)$$ p ( G ) 作为最大整数 p 使得 $$\mathbb {Z}^p$$ Z p 同构于 a G 和 $$\mathrm {q}(G)$$ q ( G ) 的离散子群是最大整数 q 使得 $$\mathbb {R}^q$$ R q 同构于 G 的闭子群。本文的目的是研究这两个不变量的性质。我们将证明如果 G 是一个非紧连通李群，则 $$1\le \mathrm {q}(G)\le \mathrm {p}(G)\le \dim (G/K)$$ 1 ≤ q ( G ) ≤ p ( G ) ≤ dim ( G / K ) 其中 K 是 G 的极大紧子群。在 G 是指数李群或 G 是连通幂零李群的情况下，我们给出了这两个不变量和众所周知的李代数不变量 $$\mathcal M(\mathfrak {g})$$ M ( g) , 即