At first, if a group G is an infinite loop space (all are based), then `\pi_0(G)` must be Abelian. Therefore, if G is discret, then it must be Abelian. In fact, any Abelian group does be infinite loop space, by the EM space construction. But we have non-Abelian examples, the infinite groups U and O are infinite loop spaces, by Bott periodicity. (Does this contradict with the statement that the coefficient of a cohomology must be an Abelian group?) Are there any other examples? 



Which Groups are Infinite Loop Spaces?
Edit: I just reread the question, and it says "*if* a group is an infinite loop space..." I realise the first paragraph of my answer is incorrect. The rest still stands.




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Firstly, $\pi\_0(G)$ does not have to be abelian - it is $\pi\_1(G,e)$ which is abelian, as the Eckmann-Hilton argument shows.


The coefficients of cohomology do not *have* to be abelian groups, but I guess you are referring to the idea that the ordinary cohomology of a space (singular, or sheaf, say) only makes sense in all non-negative dimensions for abelian-group coefficients. One can define $H^1(X,G)$ ($X$ a space) for a nonabelian (topological) group, but for higher degree cohomology it is not straightforward, see [this MO answer](https://mathoverflow.net/questions/36466/do-we-have-non-abelian-sheaf-cohomology/36508#36508).


Now notice that one can use loop spectra as coefficients for cohomology, but one gets an extraordary cohomology theory: this is the case of $U$ and $O$, which represent spectra, and given $K$-theory and $KO$-theory respectively. (see the Wikipedia page on K-theory for example)


