For x an element in G prove that x and x1 have the same orde

For x an element in G prove that x and x^-1 have the same order.

Solution

Let order of x = m and order of x^-1 = n

i.e. x^m = e and (x^-1)ⁿ = e

We have to prove that m = n

  x^m (x^-1)ⁿ = e.e

   x^m x^-n = e.e

x^m-n = e

It is possible only when,

m-n = 0

Hence, m= n

Therefore,

order of x = order of x^-1