We know that there is a fiber sequence:
$$
 ... \to B^3 Z \to B String \to B Spin \to B^2 Z \to ...
$$

Is this fiber sequence induced from a short exact sequence? 

- If so, is that
$$
1 \to B^2 Z = B S^1= CP^{\infty} \to  String \to  Spin \to 1?
$$

If so, is the String group contains a normal subgroup $B^2 Z = B S^1= CP^{\infty}$. 

- The classifying space $B S^1$ of the abelian group $S^1$ is also a group? Is $CP^{\infty}$ an abelian group or nonabelian group?

- So $CP^{\infty}$ is a normal subgroup of $String$, so 
$String/CP^{\infty}=Spin$ where $Spin$ is a quotient group of $String$ group?


Please kindly correct me if I said anything wrong or stupid! Many thanks(giving)!