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!