Regarding the map L : G -> Sym(G/H),    g -> L_g given by L_g(xH) = gxH, we need to show the following.

For each g G, the image L_g is a bijection G/H -> G/H.

The map L is a morphism of groups.

The permutation representation L is transitive.

The stabilizer of the element H of G/H coincides with H.

As, for each x G,

L_g^-1(L_g(xH)) = g^-1(g(xH)) = xH,

we have (L_g)^-1 = L_g^-1.

Let g, h G. We need to show L_g · L_h = L_gh.

For each kH G/H we have

L_gh(kH) = ghkH = L_g(hkH) = L_g(L_h(kH)) = (L_g · L_h)(kH).

So the map L is indeed a morphism.

[In other words, G -> Sym(G/H) is a permutation representation.]

The stabilizer is K = {k G | kH = H}.

If k K, then there are h₁, h₂ H with kh₁ = h₂, and so k = h₂h₁^-1 H, proving K H.

Conversely, if h H, then hH = H, so h K, proving H K. Hence H = K.