Suppose $G$ is an elementary abelian $p$-group of rank n (for simplicity we can assume n=1). Denote by $\beta$ the well-known Bockstein boundary map from $H^1(G,\mathbb F\_p)$ to $H^2(G,\mathbb F\_p)$. I am looking for an explicit formula for $\beta(f)$ on $[g|h]$ if we know the value of $f$ on $G$. 



Explicit formula for Bockstein hom in group cohomology of elementary abelian p-groups
Write $G=\langle \sigma\rangle\cong \mathbb{Z}/p$. $f\in H^1(G,\mathbb{F}\_p)$ can be taken as group homomorphism $f: G \to \mathbb{F}\_p$. If $B$ denotes the bar resolution then $\beta(f)$ is represented by the cocycle 
$$B\_2 \to \mathbb{F}\_p,\;\; [\sigma^i,\sigma^j] \mapsto 
\begin{cases}f(\sigma) & , & i+j\ge p \newline 0 &,& i+j < p\end{cases}\qquad (0\le i,j < p)$$ 


For, $\beta(f)$ is the composition of the connecting homomorphism $\delta: H^1(G,\mathbb{F}\_p) \to H^2(G,\mathbb{Z})$ that results from the short exact sequence $0 \to \mathbb{Z} \xrightarrow{\cdot p}\mathbb{Z}\to \mathbb{F}\_p\to 0$ and the mod-p reduction $\rho: H^2(G,\mathbb{Z}) \to H^2(G,\mathbb{F}\_p)$. 


Let $f(\sigma)=k \mod p$. $\tilde{f}: B\_1 \to \mathbb{Z},\;[\sigma^i]\mapsto ik\;(0\le i< p)$ is a lift of $f$ and the homomorphism $B\_2 \xrightarrow{\partial}B\_1 \xrightarrow{\tilde{f}}\mathbb{Z}$ is 
$$[\sigma^i,\sigma^j] \mapsto \begin{cases}pk & , & i+j \ge p \newline 0 &,& i+j< p\end{cases}\qquad (0\le i,j < p)$$
Since $\delta(f)$ is represented by the cocycle $\frac{1}{p}(\tilde{f}\circ \partial)$, the explicit formula above follows. 


