# Element structure of symmetric group:S3

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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:

Element $()$ $(1, 2)$ $(2, 3)$ $(1, 3)$ $(1, 2, 3)$ $(1, 3, 2)$
$()$ $()$ $(1, 2)$ $(2, 3)$ $(1, 3)$ $(1, 2, 3)$ $(1, 3, 2)$
$(1, 2)$ $(1, 2)$ $()$ $(1, 2, 3)$ $(1, 3, 2)$ $(2, 3)$ $(1, 3)$
$(2, 3)$ $(2, 3)$ $(1, 3, 2)$ $()$ $(1, 2, 3)$ $(1, 3)$ $(1, 2)$
$(1, 3)$ $(1, 3)$ $(1, 2, 3)$ $(1, 3, 2)$ $()$ $(1, 2)$ $(2, 3)$
$(1, 2, 3)$ $(1, 2, 3)$ $(1, 3)$ $(1, 2)$ $(2, 3)$ $(1, 3, 2)$ $()$
$(1, 3, 2)$ $(1, 3, 2)$ $(2, 3)$ $(1, 3)$ $(1, 2)$ $()$ $(1, 2, 3)$

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.

## Summary

Item Value
order of the whole group (total number of elements) 6
See element structure of symmetric group:S3#Order computation
conjugacy class sizes 1,2,3
maximum: 3, number of conjugacy classes: 3, lcm: 6
number of conjugacy classes 3
See element structure of symmetric group:S3#Number of conjugacy classes
order statistics 1 of order 1, 3 of order 2, 2 of order 3
maximum: 3, lcm (exponent of the whole group): 6

## Family contexts

Family name Parameter values Information on element structure of family
symmetric group degree $n = 3$, i.e., $S_3$ element structure of symmetric groups
general linear group of degree two (see note below) prime power $q = 2$, i.e., field:F2: The group is $GL(2,2)$ element structure of general linear group of degree two over a finite field
dihedral group degree $n = 3$, order $2n = 6$, i.e., the group $D_6$ element structure of dihedral groups
general affine group of degree one prime power $q = 3$, i.e., field:F3 element structure of general affine group of degree one over a finite field

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.

## Elements

### Multiple ways of describing permutations

Cycle decomposition notation (cycles for fixed points are omitted) One-line notation, i.e., image of string $1,2,3$ Order Matrix (left action convention) Matrix (right action convention) Comment
$()$ 123 1 $\begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \\\end{pmatrix}$ $\begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \\\end{pmatrix}$ The matrices are the same for the identity element.
$(1,2)$ 213 2 $\begin{pmatrix} 0 & 1 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 1 \\\end{pmatrix}$ $\begin{pmatrix} 0 & 1 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 1 \\\end{pmatrix}$ The matrices are the same, since the element has order 2.
$(2,3)$ 132 2 $\begin{pmatrix} 1 & 0 & 0 \\ 0 & 0 & 1 \\ 0 & 1 & 0 \\\end{pmatrix}$ $\begin{pmatrix} 1 & 0 & 0 \\ 0 & 0 & 1 \\ 0 & 1 & 0 \\\end{pmatrix}$ The matrices are the same, since the element has order 2.
$(1,3)$ 321 2 $\begin{pmatrix} 0 & 0 & 1\\0 & 1 & 0 \\1 & 0 & 0\\\end{pmatrix}$ $\begin{pmatrix} 0 & 0 & 1\\0 & 1 & 0 \\1 & 0 & 0\\\end{pmatrix}$ The matrices are the same, since the element has order 2.
$(1,2,3)$ 231 3 $\begin{pmatrix} 0 & 0 & 1 \\ 1 & 0 & 0 \\ 0 & 1 & 0 \\\end{pmatrix}$ $\begin{pmatrix} 0 & 1 & 0 \\ 0 & 0 & 1 \\ 1 & 0 & 0\\\end{pmatrix}$ The matrices are transposes of each other, and are not equal to each other, since the element does not have order 2.
$(1,3,2)$ 312 3 $\begin{pmatrix} 0 & 1 & 0 \\ 0 & 0 & 1 \\ 1 & 0 & 0\\\end{pmatrix}$ $\begin{pmatrix} 0 & 0 & 1 \\ 1 & 0 & 0 \\ 0 & 1 & 0 \\\end{pmatrix}$ The matrices are transposes of each other, and are not equal to each other, since the element does not have order 2.

Here is the multiplication table using the one-line notation:

Element 123 213 132 321 231 312
123 123 213 132 321 231 312
213 213 123 231 312 132 321
132 132 312 123 231 321 213
321 321 231 312 123 213 132
231 231 321 213 132 312 123
312 312 132 321 213 123 231

### 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:

Family Parameter values Formula for order of a group in the family Proof or justification of formula Evaluation at parameter values Full interpretation of conjugacy class structure
symmetric group $S_n$ of degree $n$ degree $n = 3$ $n!$ See symmetric group, element structure of symmetric groups $3! = 3 \cdot 2 \cdot 1 = 6$ #Interpretation as symmetric group
general linear group of degree two $GL(2,q)$ over a finite field of size $q$ $q = 2$, i.e., field:F2 $(q^2 - 1)(q^2 - q)$ order formulas for linear groups of degree two. See also element structure of general linear group of degree two over a finite field $(2^2 - 1)(2^2 - 2) = (3)(2) = 6$ #Interpretation as general linear group of degree two
dihedral group $D_{2n}$ of degree $n$, order $2n$ $n = 3$ $2n$ By definition. See also element structure of dihedral groups $2n = 6$ #Interpretation as dihedral group
general affine group of degree one $GA(1,q)$ over a finite field of size $q$ $q = 3$, i.e., field:F3 $q(q - 1)$ element structure of general affine group of degree one over a finite field. $q(q - 1) = 3(3 - 1) = 3(2) = 6$ #Interpretation as general affine group of degree one
general semilinear group of degree one $\Gamma L(1,q)$ over a finite field of size $q = p^r$, $p$ prime $p = 2, r = 2, q = 4$, i.e., field:F4 $r(q - 1)$ order formulas for linear gorups of degree one, see also element structure of general semilinear group of degree one over a finite field $2(4 - 1) = 2(3) = 6$ #Interpretation as general semilinear group of degree one

## Other operations induced by group multiplication

### 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 $h$.

$()$ $(1,2)$ $(2,3)$ $(1,3)$ $(1,2,3)$ $(1,3,2)$
$()$ $()$ $()$ $()$ $()$ $()$ $()$
$(1,2)$ $(1,2)$ $(1,2)$ $(1,3)$ $(2,3)$ $(2,3)$ $(1,3)$
$(2,3)$ $(2,3)$ $(1,3)$ $(2,3)$ $(1,2)$ $(1,3)$ $(1,2)$
$(1,3)$ $(1,3)$ $(2,3)$ $(1,2)$ $(1,3)$ $(1,2)$ $(2,3)$
$(1,2,3)$ $(1,2,3)$ $(1,3,2)$ $(1,3,2)$ $(1,3,2)$ $(1,2,3)$ $(1,2,3)$
$(1,3,2)$ $(1,3,2)$ $(1,2,3)$ $(1,2,3)$ $(1,2,3)$ $(1,3,2)$ $(1,3,2)$
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. PLACEHOLDER FOR INFORMATION TO BE FILLED IN: [SHOW MORE]

### 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 $[g,h] = [h,g]^{-1}$:

$()$ $(1,2)$ $(2,3)$ $(1,3)$ $(1,2,3)$ $(1,3,2)$
$()$ $()$ $()$ $()$ $()$ $()$ $()$
$(1,2)$ $()$ $()$ $(1,3,2)$ $(1,2,3)$ $(1,2,3)$ $(1,3,2)$
$(2,3)$ $()$ $(1,2,3)$ $()$ $(1,3,2)$ $(1,2,3)$ $(1,3,2)$
$(1,3)$ $()$ $(1,3,2)$ $(1,2,3)$ $()$ $(1,2,3)$ $(1,3,2)$
$(1,2,3)$ $()$ $(1,3,2)$ $(1,3,2)$ $(1,3,2)$ $()$ $()$
$(1,3,2)$ $()$ $(1,2,3)$ $(1,2,3)$ $(1,2,3)$ $()$ $()$
The corresponding table with the right definition: PLACEHOLDER FOR INFORMATION TO BE FILLED IN: [SHOW MORE]

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

Conjugacy class (indexing partition) Elements Number of occurrences of each as commutator Probability of each occurring as the commutator of elements picked uniformly at random Total number of occurrences as commutator Total probability Explanation
1 + 1 + 1 $()$ 18 1/2 18 1/2 See commuting fraction and its relationship with the number of conjugacy classes.
2 + 1 $(1,2), (2,3), (1,3)$ 0 0 0 0 Not in the derived subgroup.
3 $(1,2,3), (1,3,2)$ 9 1/4 18 1/2

## Conjugacy class structure

FACTS TO CHECK AGAINST FOR CONJUGACY CLASS SIZES AND STRUCTURE:
Divisibility facts: size of conjugacy class divides order of group | size of conjugacy class divides index of center | size of conjugacy class equals index of centralizer
Bounding facts: size of conjugacy class is bounded by order of derived subgroup
Counting facts: number of conjugacy classes equals number of irreducible representations | class equation of a group

### Interpretation as symmetric group

FACTS TO CHECK AGAINST SPECIFICALLY FOR SYMMETRIC GROUPS AND ALTERNATING GROUPS:
Conjugacy class parametrization: cycle type determines conjugacy class (in symmetric group)
Conjugacy class sizes: conjugacy class size formula in symmetric group
Other facts: even permutation (definition) -- the alternating group is the set of even permutations | splitting criterion for conjugacy classes in the alternating group (from symmetric group)| criterion for element of alternating group to be real

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:

Partition Partition in grouped form Verbal description of cycle type Elements with the cycle type in cycle decomposition notation Elements with the cycle type in one-line notation Size of conjugacy class Formula calculating size Even or odd? If even, splits? Order
1 + 1 + 1 1 (3 times) three fixed points $()$ -- the identity element 123 1 $\! \frac{3!}{(1)^3(3!)}$ even; no 1
2 + 1 2 (1 time), 1 (1 time) transposition: one 2-cycle, one fixed point $(1,2)$, $(1,3)$, $(2,3)$ 213, 321, 132 3 $\! \frac{3!}{(2)(1)}$ odd 2
3 3 (1 time) one 3-cycle $(1,2,3)$, $(1,3,2)$ 231, 312 2 $\! \frac{3!}{3}$ even; no 3
Total (3 rows -- 3 being the number of unordered integer partitions of 3) -- -- -- -- 6 (equal 3!, the size of the symmetric group) -- odd: 3 and even;no: 3 order 1: 1, order 2: 3, order 3: 2

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.

FACTS TO CHECK AGAINST ON FIXED POINTS AND CYCLES
Fixed points: probability distribution of number of fixed points of permutations | expected number of fixed points of permutation equals one
Number of cycles: probability distribution of number of cycles of permutations | expected number of cycles of permutation equals harmonic number of degree
Partition Number of elements in conjugacy class Order of elements Number of fixed points Number of cycles (including fixed points) Minimum number of transpositions that must be multiplied to obtain this cycle decomposition
1 + 1 + 1 1 1 3 3 0
2 + 1 3 2 1 2 1
3 2 3 0 1 2
Mean over conjugacy classes 2 2 4/3 2 1
Mean over elements 7/3 13/6 1 11/6 7/6

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

Compare with element structure of general linear group of degree two#Conjugacy class structure

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

Nature of conjugacy class Eigenvalues Characteristic polynomial Minimal polynomial Size of conjugacy class (generic $q$) Size of conjugacy class ($q = 2$) Number of such conjugacy classes (generic $q$) Number of conjugacy classes ($q = 2$) Total number of elements (generic $q$) Total number of elements ($q = 2$) Representative as permutation
Diagonalizable over $\mathbb{F}_q$ (here $\mathbb{F}_2$) with equal diagonal entries, hence a scalar $\{1,1\}$ $(x - 1)^2 = x^2 + 1$ $x - 1$ 1 1 $q - 1$ 1 $q - 1$ 1 $()$
Diagonalizable over $\mathbb{F}_{q^2}$ (here $\mathbb{F}_4$), not over $\mathbb{F}_q$ (here $\mathbb{F}_2$). Must necessarily have no repeated eigenvalues. Pair of conjugate elements of $\mathbb{F}_{4}$ $x^2 + x + 1$, irreducible Same as characteristic polynomial $q(q - 1)$ 2 $q(q - 1)/2$ 1 $q^2(q - 1)^2/2$ 2 $(1,2,3)$
Not diagonal, has Jordan block of size two $1$ (multiplicity two) $(x - 1)^2 = x^2 + 1$ Same as characteristic polynomial $q^2 - 1$ 3 $q - 1$ 1 $(q + 1)(q - 1)^2$ 3 $(1,2)$
Diagonalizable over $\mathbb{F}_q$ with distinct diagonal entries -- -- -- $q(q + 1)$ 6 $(q - 1)(q - 2)/2$ 0 $q(q + 1)(q - 1)(q - 2)/2$ 0 --
Total (--) -- -- -- -- -- $q^2 - 1$ 3 $q^4 - q^3 - q^2 + q$ 6 --

### Interpretation as dihedral group

Compare with element structure of dihedral groups#Odd degree case

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 $(1,2)$).

Nature of conjugacy class Size of each conjugacy class (generic odd $n$) Size of each conjugacy class ($n = 3$) Number of such conjugacy classes (generic odd $n$) Number of such conjugacy classes ($n = 3$) Total number of elements (generic odd $n$) Total number of elements ($n = 3$) Representatives as permutations
Identity element 1 1 1 1 1 1 $()$
Non-identity elements in cyclic subgroup $\langle a \rangle$, where each element and its inverse form a conjugacy class 2 2 $(n - 1)/2$ 1 $n - 1$ 2 $(1,2,3)$
Elements outside the cyclic subgroup $\langle a \rangle$, all form a single conjugacy class $n$ 3 1 1 $n$ 3 $(1,2)$
Total (--) -- -- $(n + 3)/2$ 3 $2n$ 6 --

### Interpretation as general affine group of degree one

Compare with element structure of general affine group of degree one over a finite field#Conjugacy class structure

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:

Nature of conjugacy class Size of conjugacy class (generic $q$) Size of conjugacy class ($q = 3$) Number of such conjugacy classes (generic $q$) Number of such conjugacy classes ($q = 3$) Total number of elements (generic $q$) Total number of elements ($q = 3$) Representatives of conjugacy classes as permutations
$a = 1, v = 0$ 1 1 1 1 1 1 $()$
$a = 1, v \ne 0$ (conjugacy class is independent of choice of $v$) $q - 1$ 2 1 1 $q - 1$ 2 $(1,2,3)$
$a \ne 1$ (conjugacy class is determined completely by choice of $a$ and is independent of choice of $v$; in other words, each conjugacy class is a coset of the subgroup of translations) $q$ 3 $q - 2$ 1 $q(q - 2)$ 3 $(1,2)$
Total (--) -- -- $q$ 3 $q(q - 1)$ 6 --

### Interpretation as general semilinear group of degree one

Compare with element structure of general semilinear group of degree one over a finite field#Conjugacy class structure

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 $q$.

Nature of conjugacy class Size of conjugacy class (generic $p$) Size of conjugacy class ($p = 2$) Number of such conjugacy classes (generic $p$) Number of such conjugacy classes ($p = 2$) Total number of elements (generic $p$) Total number of elements ($p = 2$) Representative as permutation (one per class)
in the multiplicative group and in the prime subfield 1 1 $p - 1$ 1 $p - 1$ 1 $()$
outside the prime subfield 2 2 $p(p - 1)/2$ 1 $p(p - 1)$ 2 $(1,2,3)$
outside the multiplicative group $\frac{p^2 - 1}{p - 1} = p + 1$ 3 $p - 1$ 1 $p^2 - 1$ 3 $(1,2)$
Total -- -- $(p - 1)(p + 4)/2 = (p^2 + 3p - 4)/2$ (equals number of conjugacy classes in the group) 3 $2(p^2 - 1)$ (equals order of the whole group) 6 --

## Conjugacy class structure: additional information

### 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:

Family Parameter values Formula for number of conjugacy classes of a group in the family Proof or justification of formula Evaluation at parameter values Full interpretation of conjugacy class structure
symmetric group $S_n$ of degree $n$ degree $n = 3$ number of unordered integer partitions of $n$ Follows from cycle type determines conjugacy class. For more, see element structure of symmetric groups number of unordered integer partitions of 3 equals 3 #Interpretation as symmetric group
general linear group of degree two $GL(2,q)$ over a finite field of size $q$ $q = 2$, i.e., field:F2 $q^2 - 1$ element structure of general linear group of degree two over a finite field. See also number of conjugacy classes in general linear group of fixed degree over a finite field is polynomial function of field size $2^2 - 1 = 3$ #Interpretation as general linear group of degree two
dihedral group $D_{2n}$ of degree $n$, order $2n$ $n = 3$ $(n + 3)/2$ for odd $n$
$(n + 6)/2$ for even $n$
element structure of dihedral groups Since 3 is odd, we use the odd degree case formula: $(3 + 3)/2 = 3$ #Interpretation as dihedral group
general affine group of degree one $GA(1,q)$ over a finite field of size $q$ $q = 3$, i.e., field:F3 $q$ element structure of general affine group of degree one over a finite field. See also number of conjugacy classes in general affine group of fixed degree over a finite field is polynomial function of field size $q = 3$ #Interpretation as general affine group of degree one
general semilinear group of degree one $\Gamma L(1,q)$ over a finite field of size $q = p^2$, $p$ prime $p = 2$ so $q = p^2 = 4$, i.e., field:F4 $(p - 1)(p + 4)/2$ element structure of general semilinear group of degree one over a finite field $(2 - 1)(2 + 4)/2 = 3$ #Interpretation as general semilinear group of degree one

### Convolution algebra on conjugacy classes

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

Partition/conjugacy class $()$ $(1,2)$ $(1,2,3)$
$()$ $()$ $(1,2)$ $(1,2,3)$
$(1,2)$ $(1,2)$ $3() + 3(1,2,3)$ $2(1,2)$
$(1,2,3)$ $(1,2,3)$ $2(1,2)$ $2() + (1,2,3)$

### 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.

Further information: symmetric groups are rational

### 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. Further information: symmetric groups on finite sets are complete

## Cayley graph

### With generating set all transpositions

Note that the left and right Cayley graphs are identical because the generating set is a conjugacy class of involutions. Also, we can unambiguously assigna direction (away from the identity) to each edge because there are no cycles of odd length, which in turn follows from the fact that all the generators are odd permutations.

## Bruhat ordering

The symmetric group of degree three can be viewed as a Coxeter group, with generators $s_1 = (1,2)$ and $s_2 = (2,3)$. The presentation is:

$\langle s_1, s_2 \mid s_1^2 = e, s_2^2 = e, (s_1s_2)^3 = e \rangle$.

We can thus consider a Bruhat ordering on the elements of the symmetric group of degree three. Note that the Bruhat ordering depends on the specific choice of transpositions we use to generate the group, which in turn depends on an implicit order of the elements $\{ 1,2,3 \}$ that the group acts on (up to reversal). Thus, the Bruhat ordering is not invariant under conjugation.

The Bruhat ordering on the symmetric group of degree three has the special feature (no longer true for higher degree) that any two elements with distinct Bruhat lengths are comparable in the order. In the Bruhat ordering, there are four levels based on Bruhat length:

Length Number of elements of that length Elements of that length Conjugacy class information for these elements
0 1 $()$ -- the identity element a single conjugacy class
1 2 $\! s_1 = (1,2)$ and $\! s_2 = (2,3)$ all the elements are conjugate but do not form a complete conjugacy class
2 2 $\! s_1s_2 = (1,2,3)$ and $\! s_2s_1 = (1,3,2)$ the elements form a single conjugacy class
3 1 $\! s_1s_2s_1 = (1,3)$ a single element, part of a conjugacy class whose other elements have length 1

The element of length $3$, is, in matrix terms, the antidiagonal matrix:

$\begin{pmatrix} 0 & 0 & 1 \\ 0 & 1 & 0 \\ 1 & 0 & 0 \\\end{pmatrix}$