Specifically, I am asked to show that if $(G,\omega)$ is a $p$-valued group of finite rank, (meaning that the associated graded group $grG$ is finitely generated as an $\mathbb F_p[t]$-module), then $grG$ is free as a graded $\mathbb F_p[t]$.
I am unsure what this question is actually asking me to do, because it sounds like it is asking me to prove the assumption?
Would it not be sufficient to just take the finite generating set, and then cast out any non-linearly independent terms, thereby giving us a set that the module is free on?
Any help clarifying the objective of this question would be very much appreciated, thank you.