I was looking through my notes for a homotopy theory course and found the following mysterious statement (*K* is of course the Eilenberg-Maclane space): 


$$H^{n+1}(K(\mathbb Z\_p,n);\mathbb Z\_p) \cong \mathbb Z\_p.$$


(This would be obvious if *n+1* were replaced with *n*. This is supposed to imply that the natural transformations $H^n(X; \mathbb Z\_p)\to H^{n+1}(X; \mathbb Z\_p)$ are all multiples of the Bockstein homomorphism).


I'm at a loss trying to understand why. Spectral sequences haven't been covered yet, so there should be some simple reason. Also, is there a way to see the Bockstein in all this?


Thank you!



The (n+1)-st cohomology of K(Z/p,n).
Universal coefficient theorem + $H\_{n+1}(K(Z\_p,n);Z)=0$. An elementary way to see the latter is that the single $n+1$ cell added to $S^n$ to kill $p$ times the generator is not a cellular cycle.


