Solution
= Solution
Conversely, if all $d(P_i)$ equal $d(G)$, the same Frattini-quotient identity gives $[P_i:P_{i+1}]=p^{d(G)}$ for every $i$. This is exactly the index definition of a <uniform pro-p group>. Thus \b[(ii)$\Rightarrow$(i)], completing the equivalence of the first two conditions.