Let a and b be elements of a finite group G.
a. Prove that a and a^(-1) have the same order.
b. Prove that a and b*a*b^(-1) have the same order.
c. Prove that a*b and b*a have the same order.
Actually induction is not necessary. To show that "x" has the same order of "y", prove o(y)<=o(x) first, and then next o(x)<=o(y). Fortunately each one of the three cases is symmetric, i.e. you don't need to do anything to prove o(x)<=o(y) since you already got o(y)<=o(x)