Element structure of symmetric group:S4

This article discusses symmetric group:S4, the symmetric group of degree four. We denote its elements as acting on the set $$\{ 1,2,3,4 \}$$, written using cycle decompositions, with composition by function composition where functions act on the left.

Since this group is a complete group (i.e., every automorphism is inner and the center is trivial), the classification of elements up to conjugacy is the same as the classification up to automorphisms. Further, since cycle type determines conjugacy class for symmetric groups, the conjugacy classes are parametrized by cycle types, which in turn are parametrized by unordered integer partitions of $$4$$.

This page concentrates on the more group-theoretic aspects of the element structure. For the more combinatorial aspects, see combinatorics of symmetric group:S4.

Family contexts
Note that if you go to the section of this article, you'll find a discussion of the conjugacy class structure with each of the below family interpretations.

Multiple ways of describing permutations
Note that the matrix for the right action is obtained by taking the transpose of the matrix for the left action. For the identity element and the elements of order 2, both matrices coincide.

Order computation
The symmetric group of degree four has order 24, with prime factorization $$24 = 2^3 \cdot 3^1 = 8 \cdot 3$$. Below are listed various methods that can be used to compute the order, all of which should give the answer 24:

Computation of prime powers in order
The prime factorization of the order is:

$$24 = 2^3 \cdot 3^1$$

Conjugacy class structure
The conjugacy class sizes are $$1,3,6,6,8$$.

Interpretation as symmetric group
For any symmetric group, cycle type determines conjugacy class, i.e., the cycle type of a permutation (which describes the sizes of the cycles in a cycle decomposition of that permutation), determines its conjugacy class. In other words, two permutations are conjugate if and only if they have the same number of cycles of each size.

The cycle types (and hence the conjugacy classes) are parametrized by partitions of the size of the set. We describe the situation for this group:

 

Here is more information on the conjugacy classes:

The mean over elements of the number of fixed points is $$1$$ for all symmetric groups on finite sets. The mean over elements of the number of cycles is $$1 + 1/2 + 1/3 + \dots + 1/n$$, which in this case is $$1 + 1/2 + 1/3 + 1/4$$.

For characters, see linear representation theory of symmetric group:S4.

Interpretation as projective general linear group of degree two
The symmetric group $$S_4$$ is isomorphic to $$PGL(2,3)$$, i.e., the projective general linear group of degree two over field:F3. Compare with element structure of projective general linear group of degree two over a finite field.

Interpretation as general affine group of degree two
The symmetric group $$S_4$$ is isomorphic to $$GA(2,2)$$, i.e., the general affine group of degree two over field:F2. Compare with element structure of general affine group of degree two over a finite field. In the table below, $$q = 2$$. The transformation is of the form $$x \mapsto Ax + v$$ where $$A \in GL(2,2)$$ and $$v \in \mathbb{F}_2^2$$.

Interpretation as general semiaffine group of degree one
We view the group as the general semiaffine group of degree one $$\Gamma A(1,q)$$ with $$q = p^2$$. Here, $$q = 4$$ and $$p = 2$$.

Number of conjugacy classes
The symmetric group of degree four has 5 conjugacy classes. Below are listed various methods that can be used to compute the number of conjugacy classes, all of which should give the answer 5:

With generating set all transpositions


Here, the generating set is the set of all transpositions. Since the generating set is a conjugacy class of involutions, the left and right Cayley graphs are identical. Further, we can unambiguously give a direction to each edge (away from the identity element) because there are no cycles of odd length, which follows from the fact that all elements of the generating set are odd permutations.

The following is some useful tabulated information about the Cayley graph. The edges to/from listed here are the edges for any representative element, not the total across the conjugacy class. Note that the sum of edges to/from in each row is $$6$$, which is the number of generators used in the generating set.

Bruhat ordering
The basic picture:



A fuller picture:



Equivalence classes up to symmetries
These are equivalence classes up to the two symmetries: flipping the $$s_i$$s and a left-to-right order reversal symmetry. Elements in the same equivalence class are in the same conjugacy class in the group and the corresponding points in the Bruhat ordering are in the same orbit under automorphisms of the graph of the Bruhat ordering.