Symmetric group S4
It has 4!=24 elements and is not abelian.
- 1 Subgroups
- 2 Lattice of subgroups
- 3 Weak order of permutations
- 4 A closer look at the Cayley table
- 5 Generators and Cayley graphs
- 5.1 Rhombicuboctahedron - Generators 4, 9 or (132), (1234)
- 5.2 Truncated cube - Generators 1, 8 or (12), (234)
- 5.3 Truncated octahedron
- 5.4 Nauru graph - Generators 1, 5, 21 or (12), (13), (14)
- 6 Bit permutations
- 7 Gray code order (Steinhaus–Johnson–Trotter algorithm)
Even permutations are white:
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 A034968. A permutation and its corresponding digit sum have the same parity.
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.
The subgroups of every group form a lattice:
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 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 between them.
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 and the lowest red vertex is the meet .
|Join and meet in permutohedra|
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:
|Number of entries in join and meet table|
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
|The Cayley table again|
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:
Generators and Cayley graphs
Rhombicuboctahedron - Generators 4, 9 or (132), (1234)
Truncated cube - Generators 1, 8 or (12), (234)
Generators 1, 9 or (12), (1234)
Generators 1, 2, 6 or (12), (23), (34)
|Cayley graph and permutohedron|
Nauru graph - Generators 1, 5, 21 or (12), (13), (14)
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
Gray code order (Steinhaus–Johnson–Trotter algorithm)
|The algorithm defines a Hamiltonian path in a Cayley graph of the symmetric group. The inverse permutations define a path in the permutohedron:|
|Permutations with green or orange background are odd. Numbers in boldface type are double-transpositions. The smaller numbers below the permutations are the inversion vectors. Red marks indicate swapped elements.|