Symmetric group:S3

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This particular group is the smallest (in terms of order): non-Abelian group

This particular group is the smallest (in terms of order): non-nilpotent group

This particular group is a finite group of order: 6

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.
• It is the special linear group $SL(2,2)$. It turns out that, because of the nature of the prime two, it is also the projective special linear group $PSL(2,2)$, the general linear group $GL(2,2)$, and the projective general linear group $PGL(2,2)$.
• It is the dihedral group of order six, viz the group of (not necessarily orientation-preserving) symmetries of the equilateral triangle.
• It is the general affine group of degree $1$ over the field of three elements, i.e., $GA(1,3)$ (sometimes also written as $AGL(1,3)$).
• It is the von Dyck group with parameters $(3,2,2)$, and in particular, is a Coxeter group. In particular, it has the presentation:

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

In the Coxeter language, this is written as:

$\langle s_{1},s_{2}\mid s_{1}^{2}=s_{2}^{2}=(s_{1}s_{2})^{3}=e\rangle$.

Multiplication table

We portray elements as permutations 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. With the opposite assumption, the row element is multiplied on the right and the column element on the left:

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

Families

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

1. The family of symmetric groups (groups of all permutations on a set): A list of all members is available at Category:Symmetric groups. On account of this, $S_{3}$ is also a Coxeter group, since all symmetric groups are Coxeter groups.
2. The family of dihedral groups: A list of all dihedral groups is available at Category:Dihedral groups. In turn, this puts it in the family of von Dyck groups: $S_{3}$ is the von Dyck group with parameters $(3,2,2)$.
3. The family of general affine groups: the symmetric group on three elements can be described as the general affine group of order $1$ on a field of three elements (denoted $GA(1,3)$): in other words, it is the semidirect product of the additive group of this field, by its multiplicative group. For a prime $p$, the general affine group $GA(1,p)$ is a group of order $p(p-1)$ obtained as a semidirect product of the additive group of order $p$ and the multiplicative group of order $(p-1)$, which is cyclic.
4. The family of general linear groups: The symmetric group is isomorphic to the general linear group of order two over the field of two elements (denoted $GL(2,2)$).

(and others).

Elements

Further information: Element structure of symmetric group:S3

Upto conjugacy

There are three conjugacy classes:

• The identity element
• The conjugacy class of transpositions whose elements are $(1,2),(2,3),(1,3)$. This conjugacy class has size three, and every element in it has order two.
• The conjugacy class of 3-cycles: $(1,2,3),(1,3,2)$. This conjugacy class has size two, and every element in it has order three.

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.

Upto 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.

Arithmetic functions

Single-valued functions

Function	Value	Explanation
order	6	$3!=6$.
exponent	6	Elements of order $2$ and $3$.
Derived length	2	Cyclic subgroup of order three is abelian, has abelian quotient.
Minimum size of generating set	2	$(1,2),(1,2,3)$
subgroup rank	2	All proper subgroups are cyclic.
max-length	2	Subgroup series going through subgroup of order two or three.
number of subgroups	6	See subgroup structure of symmetric group:S3
number of conjugacy classes	3
number of conjugacy 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

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
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

Subgroups

Lattice of subgroups of the symmetric group on three letters

Further information: Subgroup structure of symmetric group:S3

There are six subgroups:

1. The identity element is the trivial subgroup (1)
2. 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)
3. 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)
4. 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

Subgroup-defining functions

Subgroup-defining function	Subgroup type in list	Isomorphism class	Comment
Center	(1)	Trivial group	The group is centerless
Commutator subgroup	(3)	Cyclic group of order three
Frattini subgroup	(1)	Trivial group	The $3$-Sylow and $2$-Sylow are maximal and intersect trivially
Socle	(3)	Cyclic group of order three	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.

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

Other descriptions include the use of SymmetricGroup:

SymmetricGroup(3)

Or, the use of DihedralGroup:

DihedralGroup(6)