We study properties and the structure of Cartan subgroups in a connected Lie group. We obtain a characterisation of Cartan subgroups which generalises Wüstner’s structure theorem for the same. We show that Cartan subgroups are same as those of the centralisers of maximal compact subgroups of the radical. Moreover, we describe a recipe for constructing Cartan subgroups containing certain nilpotent subgroups in a connected solvable Lie group. We characterise the Cartan subgroups in the quotient group modulo a closed normal subgroup as the images of the Cartan subgroups in the ambient group. We also study the density of the images of power maps on a connected Lie group and show that the image of any k-th power map has dense image if its restriction to a closed normal subgroup and the corresponding map on the quotient group have dense images.

我们研究了一个连通李群中Cartan子群的性质和结构。我们获得了Cartan子群的特征，它概括了Wüstner的结构定理。我们表明，Cartan子群与自由基的最大紧致子群的扶正器相同。此外，我们描述了一种在可连接的李群中构建包含某些幂等子群的Cartan子群的方法。我们将商组中的Cartan子组以封闭的正常子组为模来表征为周围组中Cartan子组的图像。我们还研究了一个连通李群上的功率图图像的密度，并证明了任何k如果第三次幂图对封闭法线子组的限制以及商组上的对应图具有密集图，则第3次幂图具有密集图。