I have a question about Lie groups. Let $G$ be a finite dimensional (real or complex) Lie group and $f:G \rightarrow G$ an homomorphism of Lie groups. Edit : in the view of some counter-examples let's also suppose $G$ connected.
Q1 : Suppose that $f$ is injective, is it then surjective ?
Here are my attempts so far to adress this question. It is known that an injective Lie homomorphism is an immersion so the differential $df_g$ of $f$ in any point $g\in G$ is injective. For reasons of dimension, $df_g$ is then bijective. Thus, by the inversion theorem, $f$ is a diffeormorphism from $G$ to $f(G)\leq G$.
Moreover, it is also well known that $f(G) \simeq G/Ker(f) \simeq G/\lbrace e \rbrace \simeq G$. So we have a subgroup $f(G)$ of $G$ isomorphic to $G$. Then, we can reduce the question to :
Q2 : Let $H$ be a Lie subgroup of $G$. If $H \simeq G$, do we have $H=G$ ?
Now, maybe we can progress with some topological argument. Typically, I think of the fundamental group, that is useful to link the classification of Lie groups to the classification of Lie algebras. As $H$ is diffeomorphic to $G$, we have $\pi(H)=\pi(G)$. Hence a third question :
Q3 : Let $H$ be a Lie subgroup of $G$. If $Lie(H)=Lie(G)$ and $\pi(H)=\pi(G)$, do we have $H=G$ ?
After that, I have to admit that I don't have ideas about where to look. So if someone as an answer, a hint or a reference, that would be awesome.
In addition, I have a few "bonus" questions around this subject :
B1 : If the answer to the questions above is no, what is a counterexample ? Is there a simple hypothesis we can add so that it becomes true ?
B2 : What are the possible generalisations to all this discussion ? For example with infinite dimensional Lie groups ? With topological groups ?
B3 : In the other way around, if $f\in Hom(G,G)$ is surjective, is it injective ?
Thanks a lot to all who have read till the end and to the ones who will answer.
For the sake of having an answer, and to summarize some discussion in both the question and the comments:
If $G$ is connected and $f : G \to G$ is injective, then it is an immersion, hence a local diffeomorphism, hence open. But the only open subgroup of a connected topological group $G$ is $G$ itself, since any open neighborhood of the identity in $G$ generates $G$. Hence $f$ is surjective.
So with connectivity, the answers to Q1, Q2, Q3 are all yes, and in Q3 the extra hypothesis on fundamental groups is unnecessary.
The answer to a slightly different version of Q3 is no: you might ask whether two connected Lie groups are isomorphic if their Lie algebras and fundamental groups are isomorphic. As a counterexample, $SL_2(\mathbb{R})$ has fundamental group $\mathbb{Z}$, and hence it admits a unique connected double cover whose fundamental group is also $\mathbb{Z}$ (and which also has Lie algebra $\mathfrak{sl}_2(\mathbb{R})$), the metaplectic group $Mp_2(\mathbb{R})$. These two Lie groups are not isomorphic: the metaplectic group has no faithful finite-dimensional representations.
It's a bit trickier to find a compact counterexample, but it can still be done: $SU(2) \times SU(2)$ double covers three Lie groups, namely $SU(2) \times SO(3), SO(3) \times SU(2)$, and $SO(4)$. All of these have the same Lie algebra (namely $\mathfrak{su}(2) \times \mathfrak{su}(2)$) and fundamental group (namely $\mathbb{Z}_2$) but $SO(4)$ is not isomorphic to $SU(2) \times SO(3)$: their representation theory is different.
The answer to another slightly different version of Q3 is yes: you might ask whether a map $f : G \to H$ of connected Lie groups is an isomorphism if it induces an isomorphism on Lie algebras and on $\pi_1$. This is true and follows from covering space theory, since the first hypothesis implies that $f$ induces an isomorphism on universal covers.