# Difference between revisions of "Symmetric group:S3"

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## Definition

### Verbal definitions

The symmetric group $S_3$ can be defined in the following equivalent ways:

$\langle a,b,c \mid a^2 = b^2 = c^3 = abc = e \rangle$.

In the Coxeter language, this is written as:

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

### Multiplication table

We portray elements as permutations on the set $\{ 1,2,3 \}$ using the cycle decomposition. The row element is multiplied on the left and the column element on the right, with the assumption of functions written on the left. This means that the column element is applied first and the row element is applied next.

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 used the opposite convention (i.e., functions written on the right), the row element is to be multiplied on the right and the column element on the left.

Here is the multiplication table where we use the one-line notation for permutations, where, as in the previous multiplication table, the column permutation is applied first and then the row permutation. Thus, with the left action convention, the row element is multiplied on the left and the column element on the right:

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

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## Families

The symmetric group on three elements is part of some important families:

Generic name for family member Definition Parametrization of family Parameter value(s) for this member Other members Comments
symmetric group on finite set $S_n$ group of all permutations on a finite set by a nonnegative integer $n$, denoting size of set acted on $n = 3$, so the group is $S_3$ click here for a list
Coxeter group has a presentation of a particular form Coxeter matrix describing the presentation click here for a list symmetric groups on finite sets are Coxeter groups
dihedral group $D_{2n}$ semidirect product of a cyclic group and a two-element group acting via the inverse map by a positive integer $n$ that's half the order $n = 3$, so the group is $D_6$ click here for a list
general affine group $GA(n,F)$ semidirect product of a vector space over a field with the general linear group acting on that vector space name of field $F$, degree $n$(i.e., dimension of vector space). For a finite field, we may also write the group as $GA(n,q)$ where $q$ is the size of the field field:F3 (size $q = 3$), degree one, so the group is $GA(1,\mathbb{F}_3)$ or $GA(1,3)$ click here for a list
general linear group $GL(n,F)$ general linear group of finite degree over a finite field name of field $F$, degree $n$. $F$ may be replaced by its size $q$ in case of a finite field. field:F2 (size $q = 2$), degree two, so the group is $GL(2,\mathbb{F}_2)$ or $GL(2,2)$ click here for a list
projective general linear group $PGL(n,F)$ projective general linear group of finite degree over a finite field name of field $F$, degree $n$. $F$ may be replaced by its size $q$ in case of a finite field. field:F2 (size $q = 2$), degree two, so the group is $PGL(2,\mathbb{F}_2)$ or $PGL(2,2)$ click here for a list
special linear group $SL(n,F)$ special linear group of finite degree over a finite field name of field $F$, degree $n$. $F$ may be replaced by its size $q$ in case of a finite field. field:F2 (size $q = 2$), degree two, so the group is $SL(2,\mathbb{F}_2)$ or $SL(2,2)$ click here for a list
projective special linear group $PSL(n,F)$ projective special linear group of finite degree over a finite field name of field $F$, degree $n$. $F$ may be replaced by its size $q$ in case of a finite field. field:F2 (size $q = 2$), degree two, so the group is $PSL(2,\mathbb{F}_2)$ or $PSL(2,2)$ click here for a list
general semilinear group $\Gamma L(n,F)$ semidirect product of general linear group and automorphism group of base field name of field $F$, degree $n$. $F$ may be replaced by its size $q$ in case of a finite field. field:F4 (size $q = 4$), degree one, so the group is $\Gamma L(1,\mathbb{F}_4) = \Gamma L(1,4)$ click here for a list

## Elements

Further information: Element structure of symmetric group:S3

### Conjugacy class structure

As for any symmetric group, cycle type determines conjugacy class. The cycle types, in turn, are parametrized by the unordered integer partitions of $3$. The conjugacy classes are described below.

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 for size Even or odd? If even, splits? If splits, real in alternating group? Element order Formula calculating element order
1 + 1 + 1 1 (3 times) three fixed points $()$ -- the identity element 123 1 $\! \frac{3!}{(1)^3(3!)}$ even; no 1 $\operatorname{lcm} \{ 1, 1, 1 \}$
2 + 1 2 (1 time), 1 (1 time) transposition in symmetric group:S3: one 2-cycle, one fixed point $(1,2)$, $(1,3)$, $(2,3)$ 213, 321, 132 3 $\! \frac{3!}{(2)(1)}$ odd 2 $\operatorname{lcm} \{ 2,1 \}$
3 3 (1 time) 3-cycle in symmetric group:S3: one 3-cycle $(1,2,3)$, $(1,3,2)$ 231, 312 2 $\! \frac{3!}{3}$ even; yes; no 3 $\operatorname{lcm} \{ 3 \}$
Total (3 rows -- 3 being the number of unordered integer partitions of 3) -- -- -- -- 6 (equals 3!, the size of the symmetric group) -- odd: 3
even;no: 1
even; yes; no: 2
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.

For an interpretation of the conjugacy class structure based on the other equivalent definitions of the group, visit Element structure of symmetric group:S3#Conjugacy class structure.

### Automorphism class structure

The classification of elements upto automorphism is the same as that upto conjugation; this is because the symmetric group on three elements is a complete group: a centerless group where every automorphism is inner.

## Arithmetic functions

### Basic arithmetic functions

Function Value Similar groups Explanation
order (number of elements, equivalently, cardinality or size of underlying set) 6 groups with same order as symmetric group $\! S_n, n = 3:$, $\! n! = 3! = 3 \cdot 2 \cdot 1 = 6$
as general linear group of degree two $\! GL(2,q), q = 2:$ $\! (q^2 - 1)(q^2 - q) = (2^2 - 1)(2^2 - 2) = 6$
as general affine group of degree one $GA(1,q)$, $q = 3$: $q(q - 1) = 3 \cdot 2 = 6$
as dihedral group $D_{2n}, n = 3$: $2 \cdot 3 = 6$
as general semilinear group of degree one $\Gamma L(1,q), q = 4, q = p^r, p = 2, r = 2$: $r(q - 1) = 2(4 - 1) = 2(3) = 6$

exponent of a group 6 groups with same order and exponent of a group | groups with same exponent of a group Elements of order $2$ and $3$.
as symmetric group $\! S_n, n = 3$: $\! \operatorname{lcm} \{ 1,2, \dots, n \} = \operatorname{lcm} \{ 1,2,3 \} = 6$
As general linear group of degree two $GL(2,q), q = 2$, underlying prime $p = 2$: $p(q^2 - 1) = 2(2^2 - 1) = 2(3) = 6$
As general affine group of degree one $GA(1,q)$, $q = 3$, underlying prime $p = 3$: $p(q - 1) = 3(3 - 1) = 3(2) = 6$
as dihedral group $D_{2n}, n = 3$: $\operatorname{lcm} \{ 2,n \} = \operatorname{lcm} \{ 2, 3 \} = 6$
derived length 2 groups with same order and derived length | groups with same derived length Cyclic subgroup of order three is abelian, has abelian quotient.
minimum size of generating set 2 groups with same order and minimum size of generating set | groups with same minimum size of generating set $(1,2), (1,2,3)$
As symmetric group on a finite set: 2 (see symmetric group on a finite set is 2-generated)
subgroup rank of a group 2 groups with same order and subgroup rank of a group | groups with same subgroup rank of a group All proper subgroups are cyclic.
max-length of a group 2 groups with same order and max-length of a group | groups with same max-length of a group Subgroup series going through subgroup of order two or three.

### Arithmetic functions of an element-counting nature

Function Value Similar groups Explanation
number of conjugacy classes 3 groups with same order and number of conjugacy classes | groups with same number of conjugacy classes The three classes are the identity element, the transpositions, and the 3-cycles.
As $S_n, n =3$: number of unordered integer partitions of 3, equals 3
As $GL(2,q), q = 2$: $q^2 - 1 = 2^2 - 1 = 3$
As $D_{2n}, n = 3$: $(n + 3)/2 = (3 + 3)/2 = 3$
As $GA(1,q), q = 3$: $q = 3$
As $\Gamma L(1,p^2), p = 2$: $(p^2 + 3p - 4)/2 = (2^2 + 3 \cdot 2 - 4)/2 = 3$

For a more elaborate explanation of these formulas, see element structure of symmetric group:S3#Number of conjugacy classes
number of equivalence classes under real conjugacy 3 groups with same order and number of equivalence classes under real conjugacy | groups with same number of equivalence classes under real conjugacy Same as the number of conjugacy classes, because the group is an ambivalent group.
number of conjugacy classes of real elements 3 groups with same order and number of conjugacy classes of real elements | groups with same number of conjugacy classes of real elements Same as the number of conjugacy classes, because the group is an ambivalent group.
number of equivalence classes under rational conjugacy 3 groups with same order and number of equivalence classes under rational conjugacy | groups with same number of equivalence classes under rational conjugacy Same as the number of conjugacy classes, because the group is a rational group.
number of conjugacy classes of rational elements 3 groups with same order and number of conjugacy classes of rational elements | groups with same number of conjugacy classes of rational elements Same as the number of conjugacy classes, because the group is a rational group.

### Arithmetic functions of a subgroup-counting nature

Function Value Similar groups Explanation
number of subgroups 6 See subgroup structure of symmetric group:S3
number of conjugacy classes of subgroups 4
number of normal subgroups 3 groups with same order and number of normal subgroups | groups with same number of normal subgroups
number of automorphism classes of subgroups 4

### Lists of numerical invariants

List Value Explanation/comment
conjugacy class sizes 1,2,3 See cycle type determines conjugacy class, element structure of symmetric group:S3, element structure of symmetric groups
order statistics $1 \mapsto 1, 2 \mapsto 3, 3 \mapsto 2$
degrees of irreducible representations 1,1,2 See linear representation theory of symmetric group:S3, linear representation theory of symmetric groups
orders of subgroups 1,2,2,2,3,6 See subgroup structure of symmetric group:S3

## Endomorphisms

Further information: Endomorphism structure of symmetric group:S3

### Automorphisms

Since $S_3$ is a complete group, it is isomorphic to its automorphism group, where each element of $S_3$ acts on $S_3$ by conjugation. In fact, for $n \ne 2,6$, the symmetric group $S_n$ is a complete group. Further information: Symmetric groups on finite sets are complete

## Group properties

### Important properties

Property Satisfied? Explanation Comment
Abelian group No $(1,2)$ and $(2,3)$ don't commute Smallest non-abelian group
Nilpotent group No Centerless: The center is trivial Smallest non-nilpotent group
Metacyclic group Yes Cyclic normal subgroup of order three, cyclic quotient of order two
Supersolvable group Yes Metacyclic implies supersolvable
Solvable group Yes Metacyclic implies solvable

### Other properties

Property Satisfied? Explanation Comment
T-group Yes
Monolithic group Yes Unique minimal normal subgroup of order three
One-headed group Yes Unique maximal normal subgroup of order three
Jordan-unique group Yes There is a unique composition series
SC-group Yes Every subgroup of it is a C-group C-group means that every subgroup is permutably complemented
Rational-representation group Yes Symmetric groups are rational-representation
Rational group Yes Symmetric groups are rational Also see classification of rational dihedral groups
Ambivalent group Yes Symmetric groups are ambivalent
Complete group Yes Symmetric groups are complete, except degrees $2,6$
Frobenius group Yes Frobenius kernel is alternating group, complement is any subgroup of order two. Frobenius group on account of being $GA(1,3)$.
Camina group Yes
Z-group Yes Both the 2-Sylow subgroup (S2 in S3) and the 3-Sylow subgroup (A3 in S3) are cyclic.
Schur-trivial group Yes Schur multiplier of Z-group is trivial

## Subgroups

Further information: Subgroup structure of symmetric group:S3

### Quick summary

Item Value
Number of subgroups 6
Compared with $S_n, n = 1,2,3,4,5,\dots$: 1,2,6,30,156,1455,11300, 151221
Number of conjugacy classes of subgroups 4
Compared with $S_n, n = 1,2,3,4,5,\dots$: 1,2,4,11,19,56,96,296,554,1593
Number of automorphism classes of subgroups 4
Compared with $S_n, n = 1,2,3,4,5,\dots$: 1,2,4,11,19,37,96,296,554,1593
Isomorphism classes of Sylow subgroups and the corresponding Sylow numbers and fusion systems 2-Sylow: cyclic group:Z2, Sylow number is 3, fusion system is the trivial one
3-Sylow: cyclic group:Z3, Sylow number is 1, fusion system is non-inner fusion system for cyclic group:Z3
Hall subgroups Given that the order has only two distinct prime factors, the Hall subgroups are the whole group, trivial subgroup, and Sylow subgroups. Interestingly, all subgroups are Hall subgroups, because the order is a square-free number
maximal subgroups maximal subgroups have order 2 (S2 in S3) and 3 (A3 in S3).
normal subgroups There are three normal subgroups: the trivial subgroup, the whole group, and A3 in S3.

### Table classifying subgroups up to automorphisms

Automorphism class of subgroups List of all subgroups Isomorphism class Order of subgroups Index of subgroups Number of conjugacy classes (=1 iff automorph-conjugate subgroup) Size of each conjugacy class (=1 iff normal subgroup) Total number of subgroups (=1 iff characteristic subgroup) Isomorphism class of quotient (if exists) Note
trivial subgroup $\{ () \}$ trivial group 1 6 1 1 1 symmetric group:S3 trivial
S2 in S3 $\{ (), (1,2) \}, \{ (), (2,3) \}, \{ (), (1,3) \}$ cyclic group:Z2 2 3 1 3 3 -- 2-Sylow
A3 in S3 $\{ (), (1,2,3), (1,3,2) \}$ cyclic group:Z3 3 2 1 1 1 cyclic group:Z2 3-Sylow
whole group $\{ (), (1,2,3), (1,3,2),$
$(1,2), (1,3), (2,3) \}$
symmetric group:S3 6 1 1 1 1 trivial group
Total (4 rows) -- -- -- -- 4 -- 6 -- --

### Subgroup-defining functions and associated quotient-defining functions

Subgroup-defining function What it means Value as subgroup Value as group Order Associated quotient-defining function Value as group Order (= index of subgroup)
center elements that commute with every group element trivial subgroup trivial group 1 inner automorphism group symmetric group:S3 6
derived subgroup subgroup generated by all commutators A3 in S3: $\{ (), (1,2,3), (1,3,2) \}$ cyclic group:Z3 3 abelianization cyclic group:Z2 2
Frattini subgroup intersection of all maximal subgroups trivial subgroup trivial group 1 Frattini quotient symmetric group:S3 6
Jacobson radical intersection of all maximal normal subgroups A3 in S3: $\{ (), (1,2,3), (1,3,2) \}$ cyclic group:Z3 3  ? cyclic group:Z2 2
socle join of all minimal normal subgroups A3 in S3: $\{ (), (1,2,3), (1,3,2) \}$ cyclic group:Z3 3  ? cyclic group:Z2 2
Fitting subgroup join of all nilpotent normal subgroups A3 in S3: $\{ (), (1,2,3), (1,3,2) \}$ cyclic group:Z3 3  ? cyclic group:Z2 2
solvable radical join of all solvable normal subgroups whole group symmetric group:S3 6  ? trivial group 1
Brauer core largest odd-order normal subgroup A3 in S3: $\{ (), (1,2,3), (1,3,2) \}$ cyclic group:Z3 3  ? cyclic group:Z2 2
3-Sylow core or 3-core normal core of any 3-Sylow subgroup A3 in S3: $\{ (), (1,2,3), (1,3,2) \}$ cyclic group:Z3 3  ? cyclic group:Z2 2
3-Sylow closure normal closure of any 3-Sylow subgroup A3 in S3: $\{ (), (1,2,3), (1,3,2) \}$ cyclic group:Z3 3  ? cyclic group:Z2 2

## Linear representation theory

Further information: Linear representation theory of symmetric group:S3

### Summary

Item Value
Degrees of irreducible representations over a splitting field (and in particular over $\mathbb{C}$) 1,1,2
maximum: 2, lcm: 2, number: 3
sum of squares: 6, quasirandom degree: 1
Schur index values of irreducible representations 1,1,1
Smallest ring of realization for all irreducible representations (characteristic zero) $\mathbb{Z}$
Minimal splitting field, i.e., smallest field of realization for all irreducible representations (characteristic zero) $\mathbb{Q}$ (hence, it is a rational representation group)
Condition for being a splitting field for this group Any field of characteristic not two or three is a splitting field. In particular, $\mathbb{Q}$ and $\R$ are splitting fields.
Minimal splitting field in characteristic $p \ne 0,2,3$ The prime field $\mathbb{F}_p$
Smallest size splitting field field:F5, i.e., the field of five elements.

### Character table

Representation/Conjugacy class representative $()$ (identity element) -- size 1 $(1,2,3)$ (3-cycle) -- size 2 $(1,2)$ (2-transposition) -- size 3
Trivial representation 1 1 1
Sign representation 1 1 -1
Standard representation 2 -1 0

## Distinguishing features

### Smallest of its kind

• This is the unique non-abelian group of smallest order. All groups of order up to $5$, and all other groups of order $6$, are abelian.
• This is the unique non-nilpotent group of smallest order. All groups of order up to $5$, and all other groups of order $6$, are nilpotent.
• This is the unique smallest nontrivial complete group.

## GAP implementation

### Group ID

This finite group has order 6 and has ID 1 among the groups of order 6 in GAP's SmallGroup library. For context, there are groups of order 6. It can thus be defined using GAP's SmallGroup function as:

SmallGroup(6,1)

For instance, we can use the following assignment in GAP to create the group and name it $G$:

gap> G := SmallGroup(6,1);

Conversely, to check whether a given group $G$ is in fact the group we want, we can use GAP's IdGroup function:

IdGroup(G) = [6,1]

or just do:

IdGroup(G)

to have GAP output the group ID, that we can then compare to what we want.

### Other descriptions

Description Functions used
SymmetricGroup(3) SymmetricGroup
DihedralGroup(6) DihedralGroup
GL(2,2) GL