Symmetric group S4

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Cycle graph of S4

The symmetric group S4 is the group of all permutations of 4 elements.

It has 4!=24 elements and is not abelian.

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 Sloane'sA034968. 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)

Permutations of 4 elements
Cayley table of S4
See also: A closer look at the Cayley table

Subgroups[edit | edit source]

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.

Order 12[edit | edit source]

The alternating group A4 showing only the even permutations


Order 8[edit | edit source]

Dihedral group of order 8

Dihedral group of order 8

Dihedral group of order 8


Order 6[edit | edit source]

Symmetric group S3
Symmetric group S3
Symmetric group S3
Symmetric group S3

Order 4[edit | edit source]

Klein four-group
Klein four-group
Klein four-group
Klein four-group
Cyclic group Z4
Cyclic group Z4
Cyclic group Z4

Order 3[edit | edit source]

Cyclic group Z3
Cyclic group Z3
Cyclic group Z3
Cyclic group Z3

Lattice of subgroups[edit | edit source]

The subgroups of every group form a lattice:

How the six green C2 (ordered like 1, 6,   5, 14,   2, 21) are in the four S3
30 actual subgroups, which form a lattice
11 types of subgroups when grouped by colored cycle graph
Hasse diagrams of the subgroups of S4

Weak order of permutations[edit | edit source]

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.

Permutohedron[edit | edit source]

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

Permutohedron of S4
Permutations represented by their sets of inversions
Schlegel diagram of the truncated octahedron
In the Schlegel diagram, the inversions show a remarkable symmetry.
Permutations represented by matrices
Permutations represented by permuted elements,
and inversion vectors below them
(compare convex version)
The edges of the permutohedron match transpositions, i.e. permutations exchanging only two elements.

When two permutations are linked by a highlighted edge, representing one of six transpositions,
this transposition turns one permutation into the other and vice versa.

E.g. in the top permutohedron the permutations 3 and 5 are linked by a highlighted edge, representing transposition 2.
So 23=5 and 25=3.            The Function composition gf   (spoken "g of f" or "g after f")   tells, that first f is done, and then g.

This file shows the same bits like the files and above, but with a single permutohedron for each inversion.
The highlighted edges from the file above are also shown.
A vertex is red , when it's higher than such an edge - i.e. when an arrow points in its direction directly or indirectly.
The red vertices always form a compound of two hexagons and two squares.

Join and meet[edit | edit source]

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 .

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.)

Join table of the weak order of permutations
Positions of the entries in the join table

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:

This file shows the same bits like the join table above, but with a single matrix for each inversion.

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[edit | edit source]

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:

Positions of the entries in the Cayley table

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

Rows as permutations

Columns as permutations

Generators and Cayley graphs[edit | edit source]

Rhombicuboctahedron - Generators 4, 9 or (132), (1234)[edit | edit source]

Truncated cube - Generators 1, 8 or (12), (234)[edit | edit source]

Truncated octahedron[edit | edit source]

Generators 1, 9 or (12), (1234)[edit | edit source]

Generators 1, 2, 6 or (12), (23), (34)[edit | edit source]

(compare convex version)

Nauru graph - Generators 1, 5, 21 or (12), (13), (14)[edit | edit source]

Torus embedding[edit | edit source]

Sequences with the first digit omitted
Triples linked by an edge differ in only one digit.
The edge color tells by which: 1st, 2nd, 3rd

Generalized Petersen graph[edit | edit source]

Adjacency matrix[edit | edit source]

Adjacency matrix of the Nauru graph

Bit permutations[edit | edit source]

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

    Cycle graphs of S4 with permutations pn and Pn

Gray code order (Steinhaus–Johnson–Trotter algorithm)[edit | edit source]

The algorithm defines a Hamiltonian path in a Cayley graph of the symmetric group. The inverse permutations define a path in the permutohedron:
Cayley graph
Permutations form a Gray code. The swapped elements are always adjacent.
Permutations, inversion vectors and inversion sets form a Gray code.
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.