Past exam of the mathematics course of the University of Cambridge / 2007 / iii / Paper 5 / 3 / ii / 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 (ii)$\Rightarrow$(i), completing the equivalence of the first two conditions.