2.12. Let G be a group, let d _ 1 be an integer, and define a subset of G by G[d] = {g E G : gd = e}. (a) Prove that if g is in G[d], then g- is in G [d]. (b) Suppose that G is commutative. Prove that if g1 and 92 are in G[d], then their product g1 92 is in G[d]. (c) Deduce that if G is commutative, then G d] is a group. (d) Show by an example that if G is not a commutative group, then G[d] need not be a group. (Hint. Use Exercise 2.11. )Solution to Exercise 2.12. (a) For any element h of G and any positive integer n, we have (h_1)"*h\"=(h_1*h_1*---*h_1l*(h*h*'\"*hl== since there are 91 copies of h1 to cancel the n copies of h. Thus (ll1)\" is the inverse of h\