Let G be a group and let a, b E G. (a) Prove that o(ab) = o(ba). (Note that we are not assuming that a and b commute. In fact, the interesting case is the case where a and b do not compute.)) (b) Prove that o(a-lba) = o(b). %3D (c) If ab = ba and gcd(o(a), o(b)) = 1, then prove that o(ab) = o(a)o(b).

Let G be a group. Prove that (ab)1= a"b-1 for all a and b in G if and only if G is abelian

Answer in detail both of the parts

Exercise 18. (See WOE3.pg51.) Let G = R\{1}. Define an operation ∗ on G by x ∗ y = x + y − xy, for every x, y ∈ G. (i) Show that (G, ∗) is a commutative group. (ii) Find the inverse of 3 and 4. (iii) Find a solution of the equation 4 ∗ x ∗ 3 = 5 in G. Is it unique?