I have an exact sequence $$0 \rightarrow U \rightarrow V \rightarrow V/U \rightarrow 0,$$ where $U,V, V/U$ are $\mathbb{Z}_p[G]$ modules for a finite $p$-group $G$. Does it imply that the following sequence $$0 \rightarrow U[p^j] \rightarrow V[p^j] \rightarrow (V/U)[p^j] \rightarrow 0$$ is exact? If not, what are the conditions on $U$ and $V$ to make it exact?
Here $U[p^j]$ are the $p^j$ torsion elements of $U$, $j \in \mathbb{N}$. I cannot show that the natural projection map $$ V[p^j] \rightarrow (V/U)[p^j] $$ is surjective. Thanks for your help.
Let us consider the diagram:
$\require{AMScd}$ \begin{CD} 0 @>>> U @>>> V @>>> V/U @>>> 0\\ @. @Vp^jVV @Vp^jVV @Vp^jVV \\ 0 @>>> U @>>> V @>>> V/U @>>> 0 \end{CD}
There is a long exact sequence with six terms coming from the long exact sequence in homology. The homology of the "left (vertical) complex" $U\overset{p^j}\to U$ (expanded with zero objects in other degrees) in the "first $U$", the upper one in the diagram, is the kernel of $p^j$, so it is $U[p^j]$.
The cokernel is $U/p^j$, the homology taken in the position of the "lower $U$".
Same for the other vertical complexes. We get thus the "long" exact sequence: $$ 0 \to U[p^j] \to V[p^j] \to (V/U)[p^j]\ {\color{red}{\overset\delta\to}} \ U/p^j \to V/p^j \to (V/U)/p^j \to 0\ . $$ The above delta morphism captures the information to answer the OP. It cannot be said more in this generality. (A split extension or a zero target for $\delta$ would be fine...)