On normal subgroups of semisimple Lie groups (Q708724)

The article is devoted to prove the following result: Every normal subgroup of a finite-di\-men\-sion\-al real semisimple Lie group is closed in the topology of the Lie group. This is a generalization of the result by \textit{D. L. Ragozin} [Proc. Am. Math. Soc. 32, 632--633 (1972; Zbl 0231.22012)]. Whereas Ragozin imposed the hypothesis of being connected for semisimple Lie groups, the present article gives a proof removing this assumption. The proof is done by induction on the dimension of the Lie group and uses essentially the notions and properties of arc component and centre of a group.