Symmetric group S4



The symmetric group $$S_4$$ is the group of all permutations of 4 elements.

It has $$4! = 4 \cdot 3 \cdot 2 \cdot 1 =24$$ elements and is not abelian.

Even permutations are white:
 * the identity
 * eight 3- cycles
 * three double- transpositions (in bold typeface)

Odd permutations are colored:
 * six transpositions (green)
 * six 4-cycles (orange)

The small table on the left shows the permuted elements, and inversion vectors (which are reflected factorial numbers) below them. Another column shows the inversion sets, ordered like. (When a dot with the numbers i,j is marked red, than the elements on places i,j are out of their natural order.)

The digit sums of the inversion vectors (or factorial numbers) and the cardinalities of the inversion sets are equal. They form the sequence. A permutation and its corresponding digit sum have the same parity.

The big table on the right is the Cayley table of S4. It could also be given as the matrix multiplication table of the shown permutation matrices. (Compare multiplication table for S3)

Subgroups
There are 30 subgroups of S4, including the group itself and the 10 small subgroups.

Every group has as many small subgroups as neutral elements on the main diagonal: The trivial group and two-element groups Z2. These small subgroups are not counted in the following list.

Lattice of subgroups
The subgroups of every group form a lattice:

Weak order of permutations
The permutations of n elements form a lattice. A permutation may be defined by its set of inversions; and the lattice by the subset relation $$\subseteq$$ between these sets. Or a permutation my be defined by its factorial number (or inversion vector); and the lattice by the bitwise less than or equal relation $$\le$$ between them.

Permutohedron
The Hasse diagram of the weak order of permutations is the permutohedron. For the symmetric group S4 it's the truncated octahedron.





Join and meet
If one wants to have join and meet of any two permutations, one can find them in the permutohedron. The following table shows all relevant pairs of permutations. The arguments (row and column of the table) and their join and meet are shown by red vertices in the little permutohedra. The highest red vertex is always the join $$\cup$$ and the lowest red vertex is the meet $$\cap$$.

The following join table is derived from the table above. Besides the decimal enumeration, it shows also the inversion sets and factorial numbers. (The meet table is like this one, but reflected about the subdiagonal, and with all numbers replaced by their difference with 23.)



It's worth taking a look at the number of red squares in the 24 matrices above:

The table becomes more interesting, looking only at the inversion bits. This file shows the bits of every inversion in a single matrix:



One can see, that for the three inversions comparing consecutive places, the join operation is simply the union of the inversion sets. For the two inversions comparing places with one place between, there are 32 darker red fields beyond the union. For the inversion comparing the first and the last place, there are 56 darker fields beyond the union. So the union of two permutations inversion sets is always a subset to the inversion set of the permutations' join.

A closer look at the Cayley table
Every entry appears exactly one time in every row and column of the Cayley table. So the positions of the entries form 24 permutation matrices:



Rows and columns of the Cayley table match permutations of 24 elements. Below they are also represented as permutation matrices:





Bit permutations
When the permutations pn of 4 elements are applied on the reverse binary digits of the integers 0...15, they generate permutations Pn of 16 elements, which also form the symmetric group S4. Walsh permutation; bit permutation