Element structure of symmetric group:S3

This article discusses symmetric group:S3, the symmetric group of degree three. We denote its elements as acting on the set $$\{ 1,2,3 \}$$, written using cycle decompositions, with composition by function composition where functions act on the left. The multiplication table is given below. The convention followed here is that the row element is multiplied on the left and the column element is multiplied on the right. Since functions are assumed to act on the left, this implies that the column element is the permutation that operates first:

  If we assume functions to act on the right, then the multiplication table constructed must be interpreted taking the row element as the element multiplied on the right and the column element as the element multiplied on the left.

For a complete explanation of how this multiplication table can be constructed, see the survey article construction of multiplication table of symmetric group:S3.

This article focuses on the basic abstract group structure and key attributes. For more on the combinatorics that arises specifically from its being a symmetric group, see combinatorics of symmetric group:S3.

Family contexts
Note: By isomorphism between linear groups over field:F2, we obtain that all the groups $$GL(2,2)$$, $$SL(2,2)$$, $$PGL(2,2)$$, and $$PSL(2,2)$$ are isomorphic to each other, and hence to $$S_3$$. Hence, we can also study $$S_3$$ in terms of element structure of projective general linear group of degree two over a finite field, element structure of special linear group of degree two over a finite field, and element structure of projective special linear group of degree two over a finite field.

Multiple ways of describing permutations
Here is the multiplication table using the one-line notation:

 

Order computation
The symmetric group of degree three has order 6. Below are listed various methods that can be used to compute the order, all of which should give the answer 6:

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

$$6 = 2^1 \cdot 3^1$$

Self-action by conjugation


Below is the induced binary operation where the column element acts on the row element by conjugation on the left, i.e., if the row element is $$g$$ and the column element is $$h$$, the cell is filled with $$hgh^{-1}$$.

Note that the action by conjugation functions by relabeling, so conjugating an element $$g$$ by an element $$h$$ effectively replaces each element in each cycle of the cycle decomposition of $$g$$ by the image of that element under $$h$$.

Here is the right action by conjugation. Note that the behavior is the same as for the left action when the acting element has order two.



Commutator operation


Here, the two inputs are group elements $$g,h$$, and the output is the commutator. We first give the table assuming the left definition of commutator: $$[g,h] = ghg^{-1}h^{-1}$$. Here, the row element is $$g$$ and the column element is $$h$$. Note that $$[g,h] = [h,g]^{-1}$$:

The corresponding table with the right definition:

Here is the information on the number of times each element occurs as a commutator:



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:

  This group is one of three finite groups with the property that any two elements of the same order are conjugate. The other two are the cyclic group of order two and the trivial group.

Here is some more information:

Note that the mean over elements of the number of fixed points is 1 for any symmetric group on a finite set, and the average of the number of cycles is $$1 + (1/2) + \dots + (1/n)$$.

For characters, see linear representation theory of symmetric group:S3.

Interpretation as general linear group of degree two
This group is the general linear group of degree two over field:F2.

Interpretation as dihedral group
The symmetric group of degree three is isomorphic to the dihedral group $$D_6$$ of degree three and order six (i.e., it is the dihedral group $$D_{2n}$$ of order $$2n$$ where $$n = 3$$). In the table below, we denote by $$a$$ the generator of the cyclic subgroup of order three (which we could take as the permutation $$(1,2,3)$$) and by $$x$$ one of the reflections (which we could take as $$(1,2)$$).

Interpretation as general affine group of degree one
The symmetric group of degree three is isomorphic to the general affine group of degree one over field:F3. All the elements of this group are of the form:

$$x \mapsto ax + v, a \in \mathbb{F}_q^\ast, v \in \mathbb{F}_q$$

where $$q = 3$$. Below, we interpret the conjugacy classes of the group in these terms:

Interpretation as general semilinear group of degree one
The symmetric group of degree three is isomorphic to the general semilinear group of degree one over field:F4. In other words, it is the group $$\Gamma L(1,p^2)$$ for $$p = 2$$. We will denote $$p^2$$ alternately by $$q$$.

Number of conjugacy classes
The symmetric group of degree three has 3 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 3:

Convolution algebra on conjugacy classes
The convolution algebra on conjugacy classes for this group is given by:

Rational and real conjugacy classes
Since the symmetric group of degree three is a rational group and in particular an ambivalent group, the rational conjugacy classes coincide with the conjugacy classes and the real conjugacy classes also coincide with the conjugacy classes.

Action of automorphism group on conjugacy classes
Since the symmetric group of degree three is a complete group, i.e., every automorphism is inner, the automorphism group acts as the identity on the set of conjugacy classes.

Note that the symmetric group of degree $$n$$ for $$n \ne 2,6$$ is complete.