# Why is a Lie Group simply connected iff it is simple and connected?

Yesterday i heard that a connected Lie group $G$ is simply connected (i.e. $\pi_1(G)=0$) if and only if $G$ is simple as a group, i.e. $G$ has no nontrivial normal subgroup. That sounds too good to be wrong, it explains why one would call this property "simply connectedness". Could someone briefly explain why this is true or give me some reference where i could find it?

Greetings, atajh

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## 2 Answers

Unfortunately this is not true. For instance $\mathbb{R}^n$ is simply connected and, being an infinite abelian group, is very far from being simple. When $n > 1$ it is not even simple as a Lie group, i.e., it has nontrivial connected, closed normal subgroups.

Also the other direction is false: $\operatorname{PSL}_2(\mathbb{R})$ is simple as an abstract group, but not simply connected, since $\operatorname{SL}_2(\mathbb{R}) \rightarrow \operatorname{PSL}_2(\mathbb{R})$ is a degree $2$ connected covering map. (If am not mistaken, then at least for linear groups what is true is that an abstractly simple group is of adjoint type, i.e., the dual condition to being simply connected.)

Anyway, you did the right thing: if you have a healthy level of interactions with other members of the mathematical community (students, faculty, etc.) then you will "hear things" which are quite new to you. Not everything you hear is actually true -- or especially, what you heard / remembered is not actually true -- so it's important to try to verify / disprove these things that you hear, or ask someone else about them.

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Thanks a lot, to both of you. – atajh Dec 2 '11 at 10:39

It's not true. $SU(2)$ is simply connected but is not simple as a group -- the center $\{\pm I\}$ is nontrivial. (There seems to be a concept called a simple Lie group which is not just Lie groups that happen to be simple, in that it asks for connected normal subgroups, so SU(2) does qualify there).

On the other hand $SO(3)$ is not simply connected, but is a simple group.

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