0

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.

mzp
  • 1,925
  • 2
  • 18
  • 41
krishneel
  • 35
  • Start with the ones you know from S3. If 4 is fixed and you get a leftover permutation of the other 3. Same with fixing anything else. Now what are you still missing? – AHusain Oct 30 '16 at 23:28

2 Answers2

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
George Law
  • 4,103
1

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).

P Vanchinathan
  • 19,504