I have some problems understanding this presentation of the Bockstein Spectral Sequence (McCleary pg 460).
Q1) Firstly, how does this short exact sequence of coefficients work? $$0\to\mathbb{Z}/p\mathbb{Z}\to\mathbb{Z}/p^2\mathbb{Z}\to\mathbb{Z}/p\mathbb{Z}$$ I am of the impression that the $\mathbb{Z}/p\mathbb{Z}\to\mathbb{Z}/p^2\mathbb{Z}$ map is the times $p$ map. In that case, shouldn't it be $\mathbb{Z}/p\mathbb{Z}\to\mathbb{Z}/p\mathbb{Z}$ instead?
Q2) I also don't get how do we see that $d^1=\beta$, and why it is such a coincidence that $\beta$ happens to be the connecting homomorphism.
Thanks a lot for any help.
1) You're correct. The map $\mathbb{Z}_p\rightarrow \mathbb{Z}_{p^2}$ is multiplication by $p$. That is $(x\mod p)\mapsto (px\mod {p^2})$. The quotient is $\mathbb{Z}_{p^2}/\mathbb{Z}_p\cong \mathbb{Z}_p$. I'd suggest writing it explicitly for $p=2,3$ to convince yourself.
As for 2). First note the theconnecting map of the bottom sequence is $\beta$ (pg 127). Now write out (the relevant terms of) the long exact sequences induced by the two coefficient sequences, one above the other, to show that the connecting map for the bottom sequence factors $\beta:H^n(X;\mathbb{Z}_p)\xrightarrow{\Delta} H^{n+1}(X;\mathbb{Z})\xrightarrow{red} H^{n+1}(X,\mathbb{Z}_p)$. This is precisely the definition of $d^1$.