I am reading Lewis' paper "Is there a convenient category of spectra?". To prove the main result on the non-existence of such a nice category, he shows that otherwise the unit component of $QS^0= \varinjlim \Omega^n S^n$ would have to be weakly equivalent a product of Eilenberg-Mac Lane spaces. So far so good, but it isn't immediately clear to me:

**Q:** *Why can't the unit component of $QS^0$ be a product of Eilenberg-Mac Lane spaces?*

This should probably be very easy, given how the paper feels no need to include any justification for it.