I wish to know the general strategy to deal with a compact Lie group $G$ and its Lie group homomorphism $G$ to $S^1$ as: $$ \text{Hom}(G, S^1). $$ Here $S^1=U(1)$.
For example, how could we determine: $$ \text{Hom}(SO(n), S^1)=? $$ $$ \text{Hom}(SU(n), S^1)=\mathbb{Z}_n? $$ $$ \text{Hom}(Sp(n), S^1)=? $$ $$ \text{Hom}(Spin(n), S^1)=? $$ $$ \text{Hom}(F_4, S^1)=? $$ $$ \text{Hom}(G_2, S^1)=? $$ $$ \text{Hom}(E_6, S^1)=?, \text{Hom}(E_7, S^1)=?, \quad \text{Hom}(E_8, S^1)=? $$
My attempt: $$ \text{Hom}(U(1), S^1)=\mathbb{Z}. $$ $$ \text{Hom}(\mathbb{Z}_n, S^1)=\mathbb{Z}_n $$ $$ \text{Hom}(\mathbb{Z}, S^1)=? $$
The following structure also may be helpful: Wite $G_{con}$ for the connected component of the identity. Write $G_{sol}$ for the largest connected normal solvable subgroup. $G_{nil}$ for the largest connected normal nilpotent subgroup, we have a sequence of normal subgroups $ 1 \subseteq G_{nil} \subseteq G_{sol} \subseteq G_{con} \subseteq G. $
Can you determine the unfilled question marks above?