How to find the elements of $\langle (1,2,3,4) \rangle$ in $S_4$?

Question

How to find the elements of $\langle (1,2,3,4) \rangle$ in $S_4$?

The answer is given as $\{\mbox{id}, (1,2,3,4), (1,3).(2,4), (1,4,3,2)\}$.

I understand how we got the first $2$ elements. Also $(1,3).(2,4)$ is got by multiplying $(1,2,3,4)$ with itself but how do we get $(1,4,3,2)$?

Answer 1

$(1432)=(1234)^{-1}$... or equivalently $(1432)=(1234)^{3}$.

Answer 2

Firstly note that $(1234)$ is a cycle of order four. So it admits four different cycles. The first one is the identity and the other three are its powers up to its third power. Its third power also corresponds to its inverse, since it is a cycle of order four.