TITLE: Group Structure on CP^infinty
QUESTION [16 upvotes]: I was inspired by the following algebraic topology orals question:
"Is $S^1$ the loop space of another space?"
This is easy to see if you recognize that $S^1$ is a $K(\mathbb{Z},1)$, and the loop space of any $K(G,n)$ is a $K(G,n-1)$. 
I then also remembered that the loop space functor is a functor from pointed topological spaces and continuous maps to the category of H-spaces and continuous homomorphisms. H-spaces being topological spaces that satisfy the axioms of a group up to homotopy (see Spanier, Chapter 1, Section 5).
I have three questions:

Is there a useful criterion for when an H-space is actually a topological group?
Seeing that $S^1$,$S^3$, and $S^7$ are the only spheres that support group structures, it doesn't seem coincidental that $S^1$ is a loop space, because it is in fact an H-space. Since $CP^{\infty}$ is the loop space of $K(Z,3)$ it too is an H-space, but is it known if it is a topological group? 
Even if not, is there a way (other than concatenation of loops) to "see" this structure on $CP^{\infty}$?

Thanks!

REPLY [3 votes]: (Re: 2 & 3)
Since $\operatorname{Sym}^n\mathbb CP^1=\mathbb CP^n$ («by Viète's formulas»), $\mathbb CP^\infty$ is the free abelian monoid generated by $\mathbb CP^1=S^2$ (with some fixed point as a unit). (This is exatly the operation «representing» tensor product of $U(1)$-bundles aka addition in $H^2$ on $[-,\mathbb CP^\infty]=\operatorname{Bun}_{U(1)}(-)=H^2(-;\mathbb Z)$ — cf. other answers.)
Unfortunately, this operation lacks (stict) inverse. But by Dold-Thom theorem $\widetilde{\mathbb Z[S^2]}:=\mathbb Z[S^2]/\mathbb Z[pt]$ (the free abelian group generated by $S^2$ with some (fixed) point as a unit) has homotopy type of $K(\mathbb Z,2)$. Moreover, the natural map $\mathbb CP^\infty=\operatorname{Sym}^\infty(S^2)\to\widetilde{\mathbb Z[S^2]}$ is a homotopy equivalence.
(And any $K(G,n)$ can be made an abelian topological group in analogous way: $\widetilde{G[S^n]}$ has desired homotopy type.)
/* Essentially x-posted from math.SE */