Solution
= Solution
Let an order-$p^{d-1}$ <subgroup> $H$ act on $G/H$. Since the number of cosets is divisible by $p$, orbit counting and Cauchy produce a fixed coset with stabiliser extending $H$ by index $p$. Induction yields <subgroups> of every $p$-power order. Applying the same orbit count to the action of $H$ on $G/H$ leaves fixed cosets indexed by $N(H)/H$, giving $|G:H|\equiv|N(H):H|\pmod p$.