If [tex]f\inS_3[\math], show that [tex]f^6=\iota[\math].
Bear with me, I am still learning Latex.
use the $\displaystyle \Sigma$ button to input the [tex] tags correctly. you type the wrong slashes
i am pretty sure there is a theorem to this effect. but note that 6 is the LCM of the orders of all the elements of $\displaystyle S_3$. it follows that any element raised to it will give the identity.
you can look up the theorem to prove this elegantly, or since $\displaystyle S_3$ is small. just prove it by cases. $\displaystyle S_3$ has elements of orders 1,2, and 3. show that for each of these kinds of elements, the 6th power yields the identity