# When does a simple Lie group contain a nonsimple subgroup?

The group of rotations of Euclidean space in $$N$$ dimensions is the special orthogonal group $$\text{SO}(N)$$. It is simple and all its Lie subgroups are (semi)simple as well.

The conformal group of Euclidean space is simple too; it's the indefinite special orthogonal group $$\text{SO}(N+1,1)$$. But one of its subgroups is the Euclidean isometry group, also called the inhomogeneous special orthogonal group $$\text{ISO}(N)$$. This subgroup is not simple: $$\text{ISO}(N) = \mathbb R^N \rtimes \text{SO}(N)$$.

The above was just an example. In general, is there any way I can know if a simple group will or will not have nonsimple subgroups?

• I expect this should happen a lot; should it only be for Lie groups of dimension $0$ , of which finite groups are special cases (try $\mathfrak{A}_n$, it contains many nonsimple subgroups). I don't know if there's an easy criterion though. Any simple Lie group that contains a free group will do – Max Jan 22 '19 at 10:54
• Your claim "all Lie subgroups of $SO(N)$ are semisimple" is false (for $N\ge 2$). Every nontrivial connected Lie group has closed connected non-simple (abelian) subgroups. – YCor Jan 22 '19 at 18:40