Let G be a commutative group with identity element c and let

Let G be a commutative group with identity element c and let S = {a G: a^2 = e}. (That is, S is the set of self-inverses of G.) Prove that S is a subgroup of G.

Solution

Let G={e, a1,a2,a3,....,an}

e is multiplicative neutral element, e=1

S={a€G: a^2=e}, a1*a1=1

Which means a1= 1/a1 .

S={a€G: a=1/a}

So S is a sub group of G