TITLE: Group structure on Eilenberg-MacLane spaces
QUESTION [10 upvotes]: How do we put a group structure on $K(G,n)$ that makes it a topological group?
I know that  $\Omega K(G,n+1)=K(G,n)$ and since we have a product of loops this makes
 $K(G,n)$ into a H-space. But what about being a topological group?

REPLY [14 votes]: The standard classifying space functor $B$ from topological groups to
topological spaces is product preserving, so it takes abelian 
topological groups to abelian topological groups.   Start
with an abelian group  $G$  as a discrete topological group, so
a $K(G,0)$.  Apply the functor $B$ iteratively $n$ times to reach
$B^nG$, which is an abelian topological group and a $K(G,n)$.