# When is $G/H = G/K$? In particular, when are two Lie groups with isomorphic Lie algebras isomorphic?

Suppose $G_1$ and $G_2$ are Lie groups with isomorphic Lie algebras. Then from standard Lie theory we know that there is a simply connected Lie group $G$ such that $G/H_i = G_i$ where $H_i$ is a discrete subgroup of the center of $G$. I am curious if there is a nice condition on $H_1,H_2$ and how they sit in $G$ that implies $G_1$ and $G_2$ are isomorphic.

More generally I guess, when is $G/H = G/K$ for a general group $G$ and normal subgroups $H$ and $K$ (maybe having $H$ and $K$ be in the center makes things easier?).

• The condition is that there is an automorphism $f : G \to G$ such that $f(H_1)=H_2$. The idea is if $h : G_1 \to G_2$ is an isomorphism, lift it to a map $G \to G$. – Ryan Budney Nov 5 '10 at 1:45
• Thanks that's simple enough! – Eric O. Korman Nov 9 '10 at 19:08
• @Ryan: I know this is old, but I see no reason why you shouldn't post your comment as an answer. – davidlowryduda May 9 '12 at 5:01

The condition is that there is an automorphism $f:G→G$ such that $f(H_1)=H_2$. The idea is if $h:G_1→G_2$ is an isomorphism, lift it to a map $G→G$.