I'm trying to prove the following statement:
Suppose $G$ and $\tilde G$ is connected and locally path connected topological groups, and $p:\tilde G\rightarrow G$ is a covering map and a homomorphism between topological groups. Prove that: the kernel of $p$ is isomorphic to the covering transformation group $Aut(\tilde G,p)$.
Here are my attempts.Since the fundamental group of a topological group is an abelian group, we have $p_*(\pi_1(\tilde G))$ is a normal subgroup of $\pi_1(G)$. So we have the group isomorphism: $$ Aut(\tilde G,p)\cong \pi_1(G)/p_*(\pi_1(\tilde G)) $$ But how to prove the right-hand-side isomorphic to $Ker\ p$?