 By the orbit-stabilizer theorem we have |"G"ω| |"G"ω| = |"G"| for each ω ∈ Ω, and therefore using the additive p-adic valuation "νp", which counts the number of factors "p", one has "νp"(|"G"ω|) + "νp"(|"G"ω|) = "νp"(|"G"|) = "k" + "r".This means that for those ω with |"G"ω| = "pk", the ones we are looking for, one has "νp"(|"G"ω|) = "r", while for any other ω one has "νp"(|"G"ω|) > "r" (as 0 < |"G"ω| < "pk" implies "νp"(|"G"ω|) < "k").Since |Ω| is the sum of |"G"ω| over all distinct orbits "G"ω, one can show the existence of ω of the former type by showing that "νp"(|Ω|) = "r" (if none existed, that valuation would exceed "r").