TITLE: Loopspace of Eilenberg Mac Lane space
QUESTION [5 upvotes]: Is the loop space of the Eilenberg-MacLane space $K(G,1)$ dependent only on the cardinality of $G$? For instance, is the loop space of $K(\mathbb{Z}_4, 1)$ homotopy equivalent to that of $K(\mathbb{Z}_2 \times \mathbb{Z}_2, 1)$?

REPLY [3 votes]: I think it's important to note that loop spaces have more structure than just their topology. Concantenation makes them into H-spaces ($A_\infty$ spaces), and if you use the Moore loop space, we can even make them into strictly associative monoids. J.P. May's book "The Geometry of Iterated Loop Spaces" basically says that every associative monoid ($A_\infty$ space) is equivalent as an associative monoid ($A_\infty$ space) to a loop space. Thus, the homotopy theory of loop spaces is essentially the same as the homotopy theory of associative monoids ($A_\infty$ spaces). Thus, when considering loop spaces we ought to consider the monoid structure as well. The monoids $G$ and $\Omega B G$ are weakly equivalent (here $BG = K(G, 1)$). So, $\Omega B G$ is really $G$ as a monoid, and $\Omega B\mathbb Z /4$ and $\Omega B(\mathbb Z /2 \oplus \mathbb Z/2)$ are not equivalent as monoids. 
Edit: The key thing to note about maps of monoids, is that they induce maps of monoids on $\pi_0$, which any old continuous map will not do. If $\Omega B\mathbb Z /4$ and $\Omega B(\mathbb Z /2 \oplus \mathbb Z/2)$ were equivalent as monoids, then on $\pi_0$ we would get an isomorphism of monoids $\mathbb Z /4 \cong \mathbb Z/2 \oplus \mathbb Z/2$.