Vtyjkyuk

I have an exact sequence $$0 rightarrow U rightarrow V rightarrow V/U rightarrow 0,$$ where $U,V, V/U$ are $mathbbZ_p[G]$ 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 $U$, $j in mathbbN$. 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:

$requireAMScd$
beginCD
0 @>>> U @>>> V @>>> V/U @>>> 0\
@. @Vp^jVV @Vp^jVV @Vp^jVV \
0 @>>> U @>>> V @>>> V/U @>>> 0
endCD

There is a long exact sequence with six terms coming from the long exact sequence in homology. The homology of the "left (vertical) complex" $Uoversetp^jto U$ (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:
$$
0
to U[p^j]
to V[p^j]
to (V/U)[p^j]
colorredoversetdeltato U/p^j
to V/p^j
to (V/U)/p^j
to 0 .
$$
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...)