Let x be an element of a group G Show that the order of x1 i

Let x be an element of a group G. Show that the order of x^-1 is the order of x.

Solution

Let the order of x be n. Then xⁿ =e so that e = eⁿ = (xx^-1)ⁿ = xⁿ (x^-1)ⁿ = e (x^-1)ⁿ = (x^-1)ⁿ . This implies that the order of x^-1 is less than or equal to n, i.e. the order of x.

Now, let order of x^-1 be m. Then (x^-1 )^m = e so that e= e^m = (xx^-1)^m= x^m (x^-1 )^m =x^me = x^m.

This implies that the order of x is less than or equal to m, i.e. the order of x^-1.

Hence the order of x^-1 is same as the order of x.