... of H in G by G/H.For each a ∈ G,define “translation” τa: G/H → G/H by τa(xH) = axH for everyx ∈ G.Fora, b ∈ G,(τa◦ τb)(xH) = τa(τb(xH)) = τa(bxH) = a(bxH) = (ab)xH, by associativity, ... x, denoted by O(x), is the subsetof XO(x) ={gx : g ∈ G}⊆X;the stabilizer of x, denoted by Gx, is the subgroupGx={g ∈ G : gx = x}≤G.If G acts on a set X,define a relation on X by x ≡ y in ... : Sub(G;K) → Sub(G/K) by : S → S/K (it is routine to check thatif S is subgroup of G containing K, then S/K is a subgroup of G/K).To see that is injective, we begin by showing that if K...