代數學證明Prove that if G is an ...

Let set S = {x ∈ G | x^2 = e}

1 e * e = e => e ∈ S

2 If Let x,y ∈ S, then x^2 = y^2 = e

(xy)^2 = xyxy = xxyy = e * e = e

xy ∈ S

3 x^2 = e => x = x^(-1)

So, [x^(-1)]^2 = e => x^(-1) ∈ S

From (1) - (3), we conclude that S is nonempty and closed under products and inverses. So, S is a subgroup of G.