Symmetric group:S3

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Definition

Verbal definitions

The symmetric group $S_{3}$ can be defined in the following equivalent ways:

It is the symmetric group on a set of three elements, viz., the group of all permutations of a three-element set. In particular, it is a symmetric group of prime degree and symmetric group of prime power degree.
It is the dihedral group of order six (degree three), viz., the group of (not necessarily orientation-preserving) symmetries of the equilateral triangle.

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It is the special linear group of degree two $S L (2, 2)$ over the field of two elements. It turns out that, because of the nature of the prime two, it is also the projective special linear group of degree two $P S L (2, 2)$ , the general linear group of degree two $G L (2, 2)$ , and the projective general linear group of degree two $P G L (2, 2)$ .
It is the general affine group of degree $1$ over the field of three elements, i.e., $G A (1, 3)$ (sometimes also written as $A G L (1, 3)$ ).
It is the von Dyck group with parameters $(3, 2, 2)$ , and in particular, is a Coxeter group. In particular, it has the presentation:

$⟨ a, b, c ∣ a^{3} = b^{2} = c^{2} = a b c = e ⟩$ .

In the Coxeter language, this is written as:

⟨ s_{1}, s_{2} ∣ s_{1}^{2} = s_{2}^{2} = (s_{1} s_{2})^{3} = e ⟩

.

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

This article is about a basic definition in group theory. The article text may, however, contain advanced material.
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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	group of all permutations on a finite set	by a nonnegative integer, denoting size of set acted on	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	semidirect product of a cyclic group and a two-element group acting via the inverse map	by a positive integer that's half the order	3	click here for a list
general linear group	general linear group of finite degree over a finite field	name of field, degree	field:F2 (size two), degree two	click here for a list
projective general linear group	projective general linear group of finite degree over a finite field	name of field, degree	field:F2 (size two), degree two	click here for a list
special linear group	special linear group of finite degree over a finite field	name of field, degree	field:F2 (size two), degree two	click here for a list
projective special linear group	projective special linear group of finite degree over a finite field	name of field, degree	field:F2 (size two), degree two	click here for a list

Elements

Further information: Element structure of symmetric group:S3

Up to conjugacy

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	$l c m {1, 1, 1}$
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	$l c m {2, 1}$
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	$l c m {3}$
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.

Up to automorphism

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.

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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 $S_{k}, k = 3 :$ , $k! = 3! = 3 \cdot 2 \cdot 1 = 6$ as general linear group of degree two $G L (2, q), q = 2 :$ $(q^{2} - 1) (q^{2} - q) = (2^{2} - 1) (2^{2} - 2) = 6$
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 $3$ .
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	$(1, 2), (1, 2, 3)$
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.
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 an ambivalent group.

Arithmetic functions of a subgroup-counting nature

Function	Value	Similar groups
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	$1, 2, 2, 2, 3$	See subgroup structure of symmetric group:S3

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

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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 $G A (1, 3)$ .
Camina group	Yes

Subgroups

Further information: Subgroup structure of symmetric group:S3

There are six subgroups:

The identity element is the trivial subgroup (1)
S2 in S3: There are three 2-element subgroups, generated by the transpositions. These are all conjugate subgroups, and each is isomorphic to the cyclic group of order two (3)
A3 in S3: There is one 3-element subgroup, generated by a 3-cycle. This is a characteristic subgroup, and is isomorphic to the cyclic group of order three. This is, concretely, the alternating group on three letters (i.e., the group of even permutations on three letters). (1)
The whole group (1)

Normal subgroups

There are three normal subgroups: the trivial subgroup (type (1)), the three-element subgroup (type (3)), and the whole group (type (4)). It turns out that these are also the same as the characteristic subgroups and the same as the fully characteristic subgroups.

Sylow subgroups

There is a unique (normal) 3-Sylow subgroup: the 3-element subgroup (type (3) in the list).
There are three 2-Sylow subgroups: the three 2-element subgroups generated by transpositions (type (2) in the list).

The Sylow subgroups in this group enjoy two special properties:

Every element in the symmetric group lies inside one of the Sylow subgroups
Every proper nontrivial subgroup is a Sylow subgroup.

Further information: Subgroup structure of symmetric group:S3

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Subgroup-defining functions

Subgroup-defining function	Subgroup type in list	Isomorphism class	Page on subgroup	Comment
center	(1)	Trivial group	--	The group is centerless
derived subgroup	(3)	cyclic group:Z3	A3 in S3
Frattini subgroup	(1)	trivial group	--	The $3$ -Sylow and $2$ -Sylow are maximal and intersect trivially
socle	(3)	Cyclic group of order three	A3 in S3	This subgroup is the unique minimal normal subgroup, i.e.,the monolith, and the group is monolithic.

Quotient-defining functions

Quotient-defining function	Isomorphism class	Comment
Inner automorphism group	Symmetric group:S3	It is the quotient by the center, which is trivial.
Abelianization	Cyclic group of order two	It is the quotient by the commutator subgroup, which is cyclic of order three.

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.

Other associated constructs

Associated construct	Isomorphism class	Comment
automorphism group	symmetric group:S3	the group is a complete group, hence is isomorphic to its automorphism group. See also symmetric groups are complete
extended automorphism group	direct product of S3 and Z2
quasiautomorphism group	direct product of S3 and Z2
outer automorphism group	trivial group	the group is a complete group, hence its outer automorphism group is trivial.
1-automorphism group	direct product of S3 and Z2
holomorph	direct product of S3 and S3

Other information

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

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