= Solution
The divisibility in (i) is the standard dimension test for a projective modular representation. If $P$ is a projective $kG$-module and $S$ is a Sylow p-subgroup of order $p^d$, then $\operatorname{Res}_S^GP$ is projective over $kS$. The <group algebra of a p-group in characteristic p is local>, so every finitely generated projective $kS$-module is free. Consequently
$$
\boxed{p^d\mid\dim_kP.}
$$
This will apply to $P=\overline W$ in the implication (v)$\Rightarrow$(i).
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