e * e = e, so e^-1 = e.

(g * h) * (h^-1 * g^-1) = g * (h * h^-1) * g^-1 = g * e * g^-1 = g * g^-1 = e,

so (g * h)^-1 = h^-1 * g^-1.

g * g^-1 = g^-1 * g = e,

so (g^-1)^-1 = g.

Follows from the previous parts.