If a [Lie group homomorphism](Lie%20group%20homomorphisms.md), $\Phi$ is one-to-one, onto, and has a continuous inverse map, $\Phi^{-1}$, then we refer to it as a _Lie Group isomorphism_. # Isomorphic Lie Groups If there exists a [Lie group](Lie%20groups.md) [[Isomorphisms]] between two groups $\mbox{G}$ and $\mbox{H}$ then we say they are isomorphic and we write this pairing as $\mbox{G}\cong\mbox{H}.$ #MathematicalFoundations/Algebra/AbstractAlgebra/GroupTheory/Lie/LieGroups