Higher Bockstein maps in group cohomology

Question

Let $p$ be an odd prime and $n>1$. I am trying to understand why the cohomology ring $H^{\ast}(\mathbb{Z}/p^n;\mathbb{F_p})$ is given by $$\mathbb{F}_p[y]\otimes\Lambda(x),$$ with $|x|=1,|y|=2$ and $\beta_n(x)=y$, where $\beta_n:H^{1}(\mathbb{Z}/p^n;\mathbb{F_p})\to H^{2}(\mathbb{Z}/p^n;\mathbb{F_p})$ is the $n$th Bockstein map. I have seen a few abstract definitions of $\beta_n$ as a higher cohomology operation or as a higher differential in the Bockstein spectral sequence. The first questions that come to my mind are:

-Can we describe explicitly $\beta_n(x)=y$ as a $2$-cocycle?

-Is the group $E=\mathbb{Z}/p\times_{y}\mathbb{Z}/p^n$ determined by $y$ abelian?

Of course, the latter question would follow from the former.

Answer 1

The homology of $\mathbb{Z}/p^n$ with integral coefficients is computed by the chain complex $$ \dots 0 \to \mathbb{Z}\stackrel{0}{\to} \mathbb{Z}\stackrel{p^n}{\to} \mathbb{Z} \stackrel{0}{\to} \dots. $$ Let $e_i$ be the generator of the $i$-th term of this complex, and let $\overline{e_i}$ be its image in the reduction modulo $p$. Then $\delta(e_1) = p^n e_2$. The way $\beta_n$ is computed on the homology of $C/p$ for $C$ a chain complex with $p$-torsion free terms is by applying the differential $\delta$ and dividing by $p^n$. This shows that $\beta_n(\overline{e_1}) = \overline{\frac{p^ne_2}{p^n}} = \overline{e_2}$, which is the generator of $H^2(\mathbb{Z}/p^n;\mathbb{F}_p)$.
The central extension corresponding to this cocycle is $\mathbb{Z}/p^{n+1}$, and it is indeed abelian. This can be seen for example from the elementary fact that if the quotient of a group by a central subgroup is cyclic then the group is abelian; choose a lift of the generator of the quotient, it commutes with the kernel of the quotient and hence everything's commutes.