Eilenberg-Mac Lane and classifying spaces

Question

What can we say about

An Eilenberg-Mac Lane space $K(G,n)$ is a classifying space $BG$.

When it could be true?

For what kind of $G$?

For what values of $n$?

References are welcomed.

Answer 1

In general for a group $G$, we have $\Omega BG \simeq G$. Thus, $BG= K(H,n)$ if and only if $G = K(H, n-1)$. That is to say, a necessary and sufficent condition is that $G$ itself is an Eilenberg-Maclane space.