How to find all the elements of the group $S_4$

Question

I know how to find all the elements up till $S_3$ but for $S_4$ I am not sure how to do that systematically.

Answer 1

Hint: $S_4$ consists of

1. 6 4-cycles
2. 8 3-cycles
3. 6 transpositions
4. 3 disjoint products of transpositions
5. 1 identity

Answer 2

Georege Law's answer tells us how to list them by various types. There is an alternative way (based on your question).

As you have listed already all permutations of $1,2,3$, we can use that to get the permutations of $1,2,3,4$.

Write out a single permutation of 1,2,3 in a piece of paper well spaced out.

Now insert 4 at front, and then at all gaps, and then finally at the end. There are 4 possible insertion slots. Each leads to a new permutation of 4 symbols.

Example from 3 1 2

4 3 1 2; 3 4 1 2; 3 1 4 2; 3 1 2 4

Now do this for every element of $S_3$.

This method applies to listing permutations of $S_{n+1}$ from $S_n$.

This has the advantage that it can be converted into a computer program (may not be the most efficient; that is a different story).