I am studying permutation groups. I understand that a symmetric group $S_n$ is the set of all permutations on $n$ symbols. So, for $S_4$ there will be 24 elements. i can write this like different arrangements of $1,2,3,4$ so $(1 2 3 4),(1 2 4 3)$ .... there will be $24$ such arrangements. Now i have two issues :-
1) Is there a decent way of writing these cycles ? with increasing $n$ it becomes very complex to compute each element
2) for $S_3$ the book says that the elements will be $(123),(12),(13),(23),(132),I$. But i see it like $(123),(132),(321), .... $ so, is there a standard way of creating the arrangement given by the book i.e directly, instead of writing first in the second way and then expressing as product of disjoint cycles ?