Let X be a G-set and let Y ⊆ X. Let G_y = {g ∈ G| gy = y for all y ∈ Y}. Show G_y is a subgroup of G, generalizing Theorem 16.12.

Data from16.12 Theorem

Let X be a G-set. Then G_x is a subgroup of G for each x ∈ X,

Proof Let x ∈ X and let g₁, g₂ ∈ G_x. Then g₁x = x and g₂x = x. Consequently, (g₁g₂)x = g₁ (g₂x) = g₁x = x, so g₁g₂ ∈ G_x, and G_x is closed under the induced operation of G. Of course ex= x, so e E G,. If g ∈ G_x, then gx = x, so x =ex= (g^-1 g)x = g¹(gx) = g-¹ x, and consequently g^-1 ∈ G x. Thus G_x is a subgroup of G.