Element structure of symmetric group:S3

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1 Other descriptions of elements
- 1.1 Multiple ways of describing permutations
2 Conjugacy class structure
3 Cayley graph
- 3.1 With generating set all transpositions
4 Bruhat ordering
5 Young diagrams and tableaux under the Robinson-Schensted correspondence
- 5.1 Summary
- 5.2 Partition details
6 Increase/decrease patterns

This article gives specific information, namely, element structure, about a particular group, namely: symmetric group:S3.
View element structure of particular groups | View other specific information about 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:

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

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 $3$ .

Other descriptions of elements

Multiple ways of describing permutations

Cycle decomposition notation	One-line notation, i.e., image of string $1,2,3$	Matrix (left action)
$()$	$1,2,3$	$\begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \\\end{pmatrix}$
$(1,2)$	$2,1,3$	$\begin{pmatrix} 0 & 1 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 1 \\\end{pmatrix}$
$(2,3)$	$1,3,2$	$\begin{pmatrix} 1 & 0 & 0 \\ 0 & 0 & 1 \\ 0 & 1 & 0 \\\end{pmatrix}$
$(1,2,3)$	$2,3,1$	$\begin{pmatrix} 0 & 0 & 1 \\ 1 & 0 & 0 \\ 0 & 1 & 0 \\\end{pmatrix}$ (note: different matrix with right action convention)
$(1,3,2)$	$3,1,2$	$\begin{pmatrix} 0 & 1 & 0 \\ 0 & 0 & 1 \\ 1 & 0 & 0\\\end{pmatrix}$ (note: different matrix with right action convention)
$(1,3)$	$3,2,1$	$\begin{pmatrix} 0 & 0 & 1\\0 & 1 & 0 \\1 & 0 & 0\\\end{pmatrix}$

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
Counting facts: Number of conjugacy classes equals number of irreducible representations | Class equation of a group

FACTS TO CHECK AGAINST SPECIFICALLY FOR SYMMETRIC GROUPS:
Conjugacy class parametrization: cycle type determines conjugacy class
Conjugacy class sizes: conjugacy class size formula for symmetric group
Other facts: splitting criterion for conjugacy classes in the alternating group

Summary

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	Verbal description of cycle type	Elements with the cycle type	Size of conjugacy class	Formula calculating size	Even or odd? If even, splits?	Order
1 + 1 + 1	three fixed points	$()$ -- the identity element	1	$\! \frac{3!}{(1)^3(3!)}$	even; no	1
2 + 1	transposition: one 2-cycle, one fixed point	$(1,2)$ , $(1,3)$ , $(2,3)$	3	$\! \frac{3!}{(2)(1)}$	odd	2
3	one 3-cycle	$(1,2,3)$ , $(1,3,2)$	2	$\! \frac{3!}{3}$	even; no	3

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.

Conjugacy class information

Partition	Number of elements in conjugacy class	Order of elements	Number of fixed points	Number of cycles (including fixed points)
1 + 1 + 1	1	1	3	3
2 + 1	3	2	1	2
3	2	3	0	1
Mean over conjugacy classes	2	2	4/3	2
Mean over elements	7/3	13/6	1	11/6

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.

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:

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

Young diagrams and tableaux under the Robinson-Schensted correspondence

Summary

Partition	Number of Young tableaux for that shape	Hook length formula	Number of permutations (via Robinson-Schensted correspondence) equals square of number of Young tableaux	List of permutations (each permutation written using one-line notation)
1 + 1 + 1	1	$\frac{3!}{3 \cdot 2 \cdot 1}$	1	$[1,2,3]$
2 + 1	2	$\! \frac{3!}{3 \cdot 1 \cdot 1}$	4	$[1,3,2]$ , $[3,1,2]$ , $[2,1,3]$ , $[2,3,1]$
3	1	$\! \frac{3!}{3 \cdot 2 \cdot 1}$	1	$[3,2,1]$

Note that the numbers in the first column are also the degrees of irreducible representations, see linear representation theory of symmetric groups and linear representation theory of symmetric group:S3.

Partition details

Here the partition is the partition for the Young diagram under the Robinson-Schensted correspondence, not the partition for the cycle type of the permutation.

Permutation (one-line notation)	Partition (Young diagram)	Position tableau	Shape tableau
$1,2,3$	1 + 1 + 1	$\begin{pmatrix}1 \\ 2 \\ 3 \\\end{pmatrix}$	$\begin{pmatrix} 1 \\ 2 \\ 3 \\\end{pmatrix}$
$2,1,3$	2 + 1	$\begin{pmatrix}1 & 2 \\ 3 & \\\end{pmatrix}$	$\begin{pmatrix}1 & 2 \\ 3 & \\\end{pmatrix}$
$1,3,2$	2 + 1	$\begin{pmatrix}1 & 2 \\ 3 & \\\end{pmatrix}$	$\begin{pmatrix} 1 & 3 \\ 2 & \\\end{pmatrix}$
$2,3,1$	2 + 1	$\begin{pmatrix}1 & 3 \\ 2 & \\\end{pmatrix}$	$\begin{pmatrix} 1 & 3 \\ 2 & \\\end{pmatrix}$
$3,1,2$	2 + 1	$\begin{pmatrix}1 & 3 \\ 2 & \\\end{pmatrix}$	$\begin{pmatrix} 1 & 2 \\ 3 & \\\end{pmatrix}$
$3,2,1$	3	$\begin{pmatrix} 1 & 2 & 3 \\\end{pmatrix}$	$\begin{pmatrix} 1 & 2 & 3 \\\end{pmatrix}$

Increase/decrease patterns

One-line notation for permutation	Increase/decrease pattern (whether each element is greater or smaller than its predecessor; an I denotes increase, a D denotes decrease	Number of Is
1,2,3	II	2
2,1,3	DI	1
1,3,2	ID	1
2,3,1	ID	1
3,1,2	DI	1
3,2,1	DD	0

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Element structure of symmetric group:S3

From Groupprops

Contents

Other descriptions of elements

Multiple ways of describing permutations

Conjugacy class structure

Summary

Conjugacy class information

Convolution algebra on conjugacy classes

Rational and real conjugacy classes

Action of automorphism group on conjugacy classes

Cayley graph

With generating set all transpositions

Bruhat ordering

Young diagrams and tableaux under the Robinson-Schensted correspondence

Summary

Partition details

Increase/decrease patterns

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