$p$-primary component of the $G$-module "commutes" with cohomology

Question

This question is related to a claim found in J-P. Serre's Galois Cohomology.

Let $G$ be a profinite group (or finite if you like) and $A$ be a torsion discrete $G$-module.

Serre says one can easily see that we can identify the $p$-primary component $H^{q}(G,A)(p)$ of $H^{q}(G,A)$ with the group $H^{q}(G,A(p))$, where $A(p)$ denotes the $ p$-primary component of $A$.

Any hints on how to prove this? Thank you!

EDIT: The answer provided reduces this to showing $H^{q}(G,A)$ is $p$-torsion, for $A$ $p$-primary. (See the comments)

Answer 1

Since $A$ is torsion, it is the direct sum of its $\ell$-torsion components over all primes $\ell$. We thus have $H^q(G,A)\cong\bigoplus_\ell H^q(G,A(\ell))$. Since $H^q(G,A(\ell))$ is $\ell$-torsion for each $\ell$, the $p$-torsion component of $\bigoplus_\ell H^q(G,A(\ell))$ is just the summand $H^q(G,A(p))$.