Let the group is symmetric group $S_4$, that express the elements of $S_4$ in cycle notation, suppose the abelian subgroup $C$ of $S_4$ to be the cyclic subgroup $$C =\{e,(1 2 3 4),(1 3)(2 4),(1 4 3 2)\}$$ How can I checked that $C$ is generated by $\sigma = (1 2 3 4)$?
What is the inverse of $a$, if $a = (1 4 3 2)$?
I know that the cyclic group is the group that generated by one element, I try by $\sigma $, $\sigma^2 $,$\sigma ^3 $ and $\sigma^4 $ mod 4 but I didnot get the original $C$, What may I do wrong to get the original $C$?
Compose $\sigma$ by its successive powers, like so: $\sigma$, $\sigma^2=\sigma\circ\sigma$, $\sigma^3=\sigma^2\circ\sigma$, $\sigma^4=\sigma^3\circ\sigma$, etc.; you'll find that you get all elements of $C$, at which point, you can conclude that $\sigma$ generates $C$.${}^\dagger$
In the process, you will find that some power $\tau$ of $\sigma$ is an inverse of $\sigma$. Simply take $\tau$ as the inverse, by uniqueness, which you will see is the element given.
$\dagger$: In response to your edit, it appears that you have made an error in your calculations. The method we have both described is correct.