The following is from McCleary's book pg 461.
Q1) I am confused about the map $$-\times p^{r-1}:H_n(X;\mathbb{Z}/p^r\mathbb{Z})\to H_n(X;\mathbb{Z}/p^r\mathbb{Z})$$
Just one page earlier, I saw this map $-\times p: H_n(X;\mathbb{Z}/p\mathbb{Z})\to H_n(X;\mathbb{Z}/p^2\mathbb{Z})$.
Why for one map, the domain and the codomain is the same (both same powers of $p$) , while for the other map the domain and codomain are different ($H_n(X;\mathbb{Z}/p\mathbb{Z})$ and $H_n(X;\mathbb{Z}/p^2\mathbb{Z})$?
Q2) I am also confused about the map $p^{r-1}H_n(X)\xrightarrow{-\times p}p^{r-1}H_n(X)$. How does it work? If we multiply some element in $p^{r-1}H_n(X)$ by $p$, shouldn't it go to $pH_n(X)$?
Thanks for any help!
(Previous page 460)
