C_{2}

and the infinite dihedral group; the two generators need not be distinct. Put

a = x y

and

b = y

. The inversion relation gives

a^{2} = x y x y = x (y x y^{- 1}) y^{2} = x x^{- 1} = e, b^{2} = e, x = ab .

(1)

a \neq = e

, both

a, b

are involutions and generate the original group. If

a = e

, then

x = y

and the group is

C_{2}

; choose the same nonidentity element twice. Thus two elements of order two generate the group, including the degenerate case allowed here. The distinction between square equal to the identity and order of a group element exactly two is essential.