keneth r rosen discrete mathematics

Question

In a group G, how to prove that (a.b)^-1 = b^-1. a^-1  for all a,b belongs to G.

Answer 1

Lets assume (a.b)^-1 = b-¹.a^-1

Then b^-1.a^-1 should be the inverse of a.b and there product should give the identity element e...

Now multiplying a.b and b^-1.a^-1 :

(a.b).(b^-1.a^-1) = e

a.(b.b^-1).a^-1 = e ............(As the group follows the Associative Property)

a.(e).a^-1 = e

a.(e.a^-1) = e

a.a^-1 = e  ......................(e.a^-1 = a^-1 as e is the identity element)

e = e

Hence L.H.S = R.H.S

And this Property Holds for the Groups....