 For a given set ω ∈ Ω, write "G"ω for its stabilizer subgroup {"g" ∈ "G" | "g"⋅ω = ω } and "G"ω for its orbit {"g"⋅ω | "g" ∈ "G"} in Ω.The proof will show the existence of some ω ∈ Ω for which "G"ω has "pk" elements, providing the desired subgroup.This is the maximal possible size of a stabilizer subgroup "G"ω, since for any fixed element α ∈ ω ⊆ "G", the image of "G"ω under the bijective map "G" → "G" of right-multiplication by α ("g" ↦ "g"α) is contained in ω; therefore, |"G"ω| ≤ |ω| = "pk".