Symmetric group:S4

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Definition

Verbal definition

The symmetric group $S_{4}$ or $S y m (4)$ , also termed the symmetric group of degree four, is defined in the following equivalent ways:

The group of all permutations, i.e., the on a set of size four. In particular, it is a symmetric group of prime power degree.
The triangle group (not the von Dyck group, but its double) $Δ (3, 3, 2)$ . In other words, it has the presentation:

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

In particular, it is a Coxeter group.

The full tetrahedral group: The group of all (not necessarily orientation-preserving) symmetries of the regular tetrahedron. This is denoted as $T_{h}$ .
The von Dyck group with parameters $(4, 3, 2)$ . In other words, it has the presentation:

$⟨ a, b, c ∣ a^{4} = b^{3} = c^{2} = a b c = e ⟩$ .

The octahedral group or cube group: group of orientation-preserving symmetries of the cube (or equivalently, the octahedron). This is denoted as $O$ .
The projective general linear group of degree two over the field of three elements: $P G L (2, 3)$ .
The general affine group of degree two over the field of two elements: $G A (2, 2)$ .

Families

The symmetric group of degree four is part of the following important families:

The family of symmetric groups (groups of all permutations on a set). A list of all symmetric groups is available at Category:Symmetric groups. On account of being a symmetric group, $S_{4}$ is also a Coxeter group, because symmetric groups on finite sets are Coxeter groups.
The family of von Dyck groups. In fact, $S_{4}$ is a spherical von Dyck group, i.e., it occurs as a finite subgroup of $S O (3, R)$ . In particular, this makes it a Coxeter group. Further information: Classification of finite subgroups of SO(3,R)
The family of triangle groups. $S_{4}$ is a spherical triangle group, i.e., a triangle group with parameters $(3, 3, 2)$ . In particular, this makes it a Coxeter group.

Arithmetic functions

Single-valued functions

Function	Value	Explanation
order	24	$4! = 24$ .
exponent	12	Elements of order $2, 3, 4$ .
derived length	3	Derived series goes through alternating group:A4 and Klein four-group of double transpositions.
nilpotency class	--	not a nilpotent group.
Frattini length	1	Frattini-free group: intersection of maximal subgroups is trivial.
minimum size of generating set	2	$(1, 2), (1, 2, 3, 4)$
subgroup rank	2	All proper subgroups are cyclic or have generating set of size two (D8, Klein four-group, and A4)
max-length	4	Series for Sylow 2-subgroup followed by whole group, OR derived series refined at end.
composition length	4
chief length	3
number of subgroups	30
number of conjugacy classes	5
number of conjugacy classes of subgroups	11

Lists of numerical invariants

List	Value	Explanation/comment
conjugacy class sizes	$1, 3, 6, 6, 8$	See element structure of symmetric group:S4, element structure of symmetric groups, cycle type determines conjugacy class
order statistics	$1 \mapsto 1, 2 \mapsto 9, 3 \mapsto 6, 4 \mapsto 8$
degrees of irreducible representations	$1, 1, 2, 3, 3$	See linear representation theory of symmetric group:S4, linear representation theory of symmetric groups

Group properties

COMPARE AND CONTRAST: Want to know more about how this group compares with symmetric groups of other degrees? Read contrasting symmetric groups of various degrees.

Property	Satisfied	Explanation	Comment
Abelian group	No	$(1, 2)$ , $(1, 3)$ don't commute	$S_{n}$ is non-abelian, $n \geq 3$ .
Nilpotent group	No	Centerless: The center is trivial	$S_{n}$ is non-nilpotent, $n \geq 3$ .
Metacyclic group	No	No cyclic normal subgroup	$S_{n}$ is not metacyclic, $n \geq 4$ .
Supersolvable group	No	No cyclic normal subgroup	$S_{n}$ is not supersolvable, $n \geq 4$ .
Solvable group	Yes	Length three, commutator subgroup is $A_{4}$ , its commutator is Klein four-group	Largest $n$ for which $S_{n}$ is solvable.
T-group	No	Double transposition generates non-normal 2-subnormal subgroup	Only $n$ for which $S_{n}$ isn't a T-group.
HN-group	No	Double transposition generates subnormal non-hypernormalized subgroup	Only $n$ for which $S_{n}$ isn't hypernormalized.
Complete group	Yes	Centerless and every automorphism's inner	Symmetric groups are complete except the ones of degree $2, 6$ .
Monolithic group	Yes	Monolith is the Klein four-group of double transpositions	All symmetric groups are monolithic; $n = 4$ is the only case the monolith is not the alternating group.
One-headed group	Yes	The alternating group is the unique maximal normal subgroup	True for all $n > 1$ .
Group having subgroups of all orders dividing the group order	Yes	(See subgroup list)	Largest $n$ for which this is true.
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

Endomorphisms

Automorphisms

Since $S_{4}$ is a complete group, it is isomorphic to its automorphism group, where each element of $S_{4}$ acts on $S_{4}$ by conjugation.

Endomorphisms

$S_{4}$ admits four kinds of endomorphisms (that is, it admits more endomorphisms, but any endomorphism is equivalent via an automorphism to one of these four):

The endomorphism to the trivial group
The identity map
The retraction to a group of order two, given by the sign homomorphism.
The retraction to a symmetric group on three of the elements, with kernel being the Klein four-group comprising the identity element and the double transpositions. (Note that all such retractions are equivalent, and there are other equivalent endomorphisms obtained by composing such a retraction with an automorphism).

Elements

Upto conjugacy

There are five conjugacy classes, corresponding to the five possible partitions of $4$ :

$4 = 1 + 1 + 1 + 1$ , i.e., cycles of size one each: The identity element. (1)
$4 = 2 + 1 + 1$ , i.e., one cycle of size two and two fixed points: The conjugacy class of transpositions, such as $(1, 2)$ . (6)
$4 = 2 + 2$ , i.e., two cycles of size two: The conjugacy class of double transpositions. (3)
$4 = 3 + 1$ , i.e., one cycle of size three and one fixed point: The conjugacy class of 3-cycles. (4)
$4 = 4$ , i.e., one cycle of size four: The conjugacy class of 4-cycles. (6)

Subgroups

Further information: Subgroup structure of symmetric group:S4

Here is a complete list of subgroups. Note that since $S_{4}$ is a complete group, every automorphism is inner, so the classification of subgroups upto conjugacy is equivalent to the classification of subgroups upto automorphism.

The trivial subgroup. Isomorphic to trivial group.(1)
The two-element subgroup generated by a transposition, such as $(1, 2)$ . Isomorphic to cyclic group of order two.(6)
The two-element subgroup generated by a double transposition, such as $(1, 2) (3, 4)$ . Isomorphic to cyclic group of order two. (3)
The four-element subgroup generated by two disjoint transpositions, such as $⟨ (1, 2), (3, 4) ⟩$ . Isomorphic to Klein four-group. (3)
The unique four-element subgroup comprising the identity and the three double transpositions. Isomorphic to Klein four-group. (1)
The four-element subgroup spanned by a 4-cycle. Isomorphic to cyclic group of order four.(3)
The eight-element subgroup spanned by a 4-cycle and a transposition that conjugates this cycle to its inverse. Isomorphic to dihedral group of order eight. (3)
The three-element subgroup spanned by a three-cycle. Isomorphic to cyclic group of order three.(4)
The six-element subgroup comprising all permutations that fix one element. Isomorphic to symmetric group on three elements. (4)
The alternating group: the subgroup of all even permutations. Isomorphic to alternating group:A4.(1)
The whole group.(1)

Normal subgroups

The only normal subgroups of $S_{4}$ are: the trivial subgroup (type (1)), the Klein four-group (type (5) -- note that type (4) are non-normal subgroups), the alternating group $A_{4}$ (type (10)), and the whole group (type (11)). It turns out that the characteristic subgroups, the fully characteristic subgroups, the permutable subgroups, and the retraction-invariant subgroups all coincide with the normal subgroups.

Sylow subgroups

The Sylow subgroups are as follows:

The $2$ -Sylow subgroups are dihedral groups of order eight, type (7) in the list.
The $3$ -Sylow subgroups are cyclic groups of order three, type (9) in the list.

Further information: Subgroup structure of symmetric group:S4

Subgroup-defining functions

Subgroup-defining function	Subgroup type in list	Isomorphism class	Comment
Center	(1)	Trivial group	The group is centerless
Commutator subgroup	(10)	Alternating group:A4
Frattini subgroup	(1)	Trivial group	The normal cores of the maximal subgroups of orders $6$ and $8$ intersect trivially.
Socle	(5)	Klein four-group	This subgroup is the unique minimal normal subgroup, i.e., the monolith, and the group is monolithic.

Supergroups

The symmetric group $S_{4}$ is contained in higher symmetric groups, most notably the symmetric group on five elements $S_{5}$ .

Extensions

These include $G L (2, 3)$ whose inner automorphism group is $S_{4}$ (specifically $S_{4}$ is the quotient of $G L (2, 3)$ by its scalar matrices).

Implementation in GAP

Group ID

This finite group has order 24 and has ID 12 among the groups of order 24 in GAP's SmallGroup library. For context, there are groups of order 24. It can thus be defined using GAP's SmallGroup function as:

SmallGroup(24,12)

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

gap> G := SmallGroup(24,12);

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

IdGroup(G) = [24,12]

or just do:

IdGroup(G)

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

Other descriptions

The group can be defined using GAP's SymmetricGroup function as:

SymmetricGroup(4)

To construct the symmetric group on a specific set of size four, enter the set in parantheses. For instance:

SymmetricGroup([1,3,4,7])

gives the group of all permutations on ${1, 3, 4, 7}$ .