# Difference between revisions of "Symmetric group:S4"

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## Definition

The symmetric group $S_4$ or $\operatorname{Sym}(4)$, also termed the symmetric group of degree four, is defined in the following equivalent ways:

$\langle s_1, s_2, s_3 \mid s_1^2 = s_2^2 = s_3^2 = e, (s_1s_2)^3 = (s_2s_3)^3 = (s_1s_3)^2 = e \rangle$.

where $e$ is the identity element.

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:

$\langle a,b,c \mid a^4 = b^3 = c^2 = abc = e \rangle$.

### Equivalence of definitions

The following is a list of proofs of the equivalence of various definitions:

## 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 4 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
von Dyck group $\langle a,b,c \mid a^p = b^q = c^r = abc = e \rangle$ $(p,q,r)$ (4,3,2) click here for a list $S_4$ is a spherical von Dyck group, i.e., it occurs as a finite subgroup of $SO(3,\R)$. In particular, this makes it a Coxeter group. Further information: Classification of finite subgroups of SO(3,R)
triangle group $\langle s_1,s_2,s_3 \mid s_1^2 = s_2^2 = s_3^2 = (s_1s_2)^p = (s_2s_3)^q = (s_3s_1)^r = e \rangle$ $(p,q,r)$ (3,3,2) click here for a list $S_4$ is a spherical triangle group
projective general linear group projective general linear group of given degree over a given field name of field, degree field:F3 (size three), degree two

## Arithmetic functions

Want to compare and contrast arithmetic function values with other groups of the same order? Check out groups of order 24#Arithmetic functions

### Single-valued functions

Function Value Similar groups Explanation for function value
order (number of elements, equivalently, cardinality or size of underlying set) 24 groups with same order As $\! S_k, k = 4:$ $\! k! = 4! = 4 \cdot 3 \cdot 2 \cdot 1 = 24$

as $\! PGL(2,q), q = 3:$ $\! (q^2 - 1)(q^2 - q)/(q - 1) = (3^2 - 1)(3^2 - 3)/2 = 24$

As $\! GA(2,q), q = 2:$ $\! q^2 \cdot (q^2 - 1) \cdot (q^2 - q) = 2^2 \cdot (2^2 - 1)(2^2 - 2) = 4 \cdot 3 \cdot 2 = 24$
exponent 12 groups with same order and exponent | groups with same exponent elements of order $2,3,4$, their lcm is $12$
derived length 3 groups with same order and derived length | groups with same derived length Derived series goes through alternating group:A4 and Klein four-group of double transpositions.
nilpotency class -- not a nilpotent group.
Frattini length 1 groups with same order and Frattini length | groups with same Frattini length Frattini-free group: intersection of maximal subgroups is trivial.
minimum size of generating set 2 groups with same order and minimum size of generating set | groups with same minimum size of generating set $(1,2), (1,2,3,4)$; see also 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 or have generating set of size two (D8, Klein four-group, and A4)
max-length of a group 4 groups with same order and max-length of a group | groups with same max-length of a group Series for Sylow 2-subgroup followed by whole group, OR derived series refined at end.
composition length 4 groups with same order and composition length | groups with same composition length
chief length 3 groups with same order and chief length | groups with same chief length
number of subgroups 30 groups with same order and number of subgroups | groups with same number of subgroups
number of conjugacy classes 5 groups with same order and number of conjugacy classes | groups with same number of conjugacy classes as $S_k, k =4:$ the number of conjugacy classes is $\! p(k) = p(4) = 5$, where $p$ is the number of unordered integer partitions; see cycle type determines conjugacy class

as $PGL(2,q), q = 3:$ the number of conjugacy classes is $q + 2 = 3 + 2 = 5$; see projective general linear group of degree two

as $GA(2,q), q = 2:$ $\! q^2 + 1 = 2^2 + 1 = 5$
number of conjugacy classes of subgroups 11 groups with same order and number of conjugacy classes of subgroups | groups with same number of conjugacy classes of subgroups

### 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 $1 \mapsto 1, 2 \mapsto 9, 3 \mapsto 8, 4 \mapsto 6$
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, $n \ge 3$.
Nilpotent group No Centerless: The center is trivial $S_n$ is non-nilpotent, $n \ge 3$.
Metacyclic group No No cyclic normal subgroup $S_n$ is not metacyclic, $n \ge 4$.
Supersolvable group No No cyclic normal subgroup $S_n$ is not supersolvable, $n \ge 4$.
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. 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

### 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

Further information: element structure of symmetric group:S4

### Conjugacy class structure

There are five conjugacy classes, corresponding to the cycle types, because cycle type determines conjugacy class. Further, each cycle type corresponds to a partition of $4$.

Partition Partition in grouped form Verbal description of cycle type Elements with the cycle type 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 1 (4 times) four cycles of size one each, i.e., four fixed points $()$ -- the identity element 1 $\! \frac{4!}{(1)^4(4!)}$ even; no 1 $\operatorname{lcm}\{ 1,1,1,1 \}$
2 + 1 + 1 2 (1 time), 1 (2 times) one transposition (cycle of size two), two fixed points $(1,2)$, $(1,3)$, $(1,4)$, $(2,3)$, $(2,4)$, $(3,4)$ 6 $\! \frac{4!}{[(2)^1(1!)][(1)^2(2!)]}$, also $\binom{4}{2}$ odd 2 $\operatorname{lcm}\{2,1,1 \}$
2 + 2 2 (2 times) double transposition: two cycles of size two $(1,2)(3,4)$, $(1,3)(2,4)$, $(1,4)(2,3)$ 3 $\! \frac{4!}{(2)^2(2!)}$ even; no 2 $\operatorname{lcm}\{2,2 \}$
3 + 1 3 (1 time), 1 (1 time) one 3-cycle, one fixed point $(1,2,3)$, $(1,3,2)$, $(2,3,4)$, $(2,4,3)$, $(3,4,1)$, $(3,1,4)$, $(4,1,2)$, $(4,2,1)$ 8 $\! \frac{4!}{[(3)^1(1!)][(1)^1(1!)]}$ or $\! \frac{4!}{(3)(1)}$ even; yes; no 3 $\operatorname{lcm}\{3,1 \}$
4 4 (1 time) one 4-cycle, no fixed points $(1,2,3,4)$, $(1,2,4,3)$, $(1,3,2,4)$, $(1,3,4,2)$, $(1,4,2,3)$, $(1,4,3,2)$ 6 $\! \frac{4!}{(4)^1(1!)}$ or $\frac{4!}{4}$ odd 4 $\operatorname{lcm} \{ 4 \}$
Total (5 rows, 5 being the number of unordered integer partitions of 4) -- -- -- 24 (equals 4!, the order of the whole group) -- odd: 12 (2 classes)
even; no: 4 (2 classes)
even; yes; no: 8 (1 class)
order 1: 1 (1 class)
order 2: 9 (2 classes)
order 3: 8 (1 class)
order 4: 6 (1 class)
--

### Automorphism class structure

Since $S_4$ is a complete group, all its automorphisms are inner automorphisms, and, in particular, the classification of elements up to conjugacy is the same as the classification up to automorphisms.

## Subgroups

Further information: Subgroup structure of symmetric group:S4

### Quick summary

Item Value
Number of subgroups 30
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 11
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 11
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: dihedral group:D8 (order 8), Sylow number is 3, fusion system is non-inner non-simple fusion system for dihedral group:D8
3-Sylow: cyclic group:Z3, Sylow number is 4, 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
maximal subgroups maximal subgroups have order 6 (S3 in S4), 8 (D8 in S4), and 12 (A4 in S4).
normal subgroups There are four normal subgroups: the whole group, the trivial subgroup, A4 in S4, and normal V4 in S4.

### Table classifying subgroups up to automorphisms

TABLE SORTING AND INTERPRETATION: Note that the subgroups in the table below are sorted based on the powers of the prime divisors of the order, first covering the smallest prime in ascending order of powers, then powers of the next prime, then products of powers of the first two primes, then the third prime, then products of powers of the first and third, second and third, and all three primes. The rationale is to cluster together subgroups with similar prime powers in their order. The subgroups are not sorted by the magnitude of the order. To sort that way, click the sorting button for the order column. Similarly you can sort by index or by number of subgroups of the automorphism class.

Automorphism class of subgroups Representative 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) Number of subgroups (=1 iff characteristic subgroup) Isomorphism class of quotient (if exists) Subnormal depth (if subnormal) Note
trivial subgroup $\{ () \}$ trivial group 1 24 1 1 1 symmetric group:S4 1
S2 in S4 $\{ (), (1,2) \}$ cyclic group:Z2 2 12 1 6 6 -- --
subgroup generated by double transposition in S4 $\{ (), (1,2)(3,4) \}$ cyclic group:Z2 2 12 1 3 3 -- 2
Z4 in S4 $\langle (1,2,3,4) \rangle$ cyclic group:Z4 4 6 1 3 3 -- --
normal Klein four-subgroup of S4 $\{ (), (1,2)(3,4),$
$(1,3)(2,4), (1,4)(2,3) \}$
Klein four-group 4 6 1 1 1 symmetric group:S3 1 2-core
non-normal Klein four-subgroups of S4 $\langle (1,2), (3,4) \rangle$ Klein four-group 4 6 1 3 3 -- --
D8 in S4 $\langle (1,2,3,4), (1,3) \rangle$ dihedral group:D8 8 3 1 3 3 -- -- 2-Sylow, fusion system is non-inner non-simple fusion system for dihedral group:D8
A3 in S4 $\{ (), (1,2,3), (1,3,2) \}$ cyclic group:Z3 3 8 1 4 4 -- -- 3-Sylow, fusion system is non-inner fusion system for cyclic group:Z3
S3 in S4 $\langle (1,2,3), (1,2) \rangle$ symmetric group:S3 6 4 1 4 4 -- --
A4 in S4 $\langle (1,2,3), (1,2)(3,4) \rangle$ alternating group:A4 12 2 1 1 1 cyclic group:Z2 1
whole group $\langle (1,2,3,4), (1,2) \rangle$ symmetric group:S4 24 1 1 1 1 trivial group 0
Total (11 rows) -- -- -- -- 11 -- 30 -- -- --

## Subgroup-defining functions

Subgroup-defining function Subgroup type in list Isomorphism class Comment
Center trivial subgroup trivial group The group is centerless
Derived subgroup A4 in S4 alternating group:A4
Frattini subgroup trivial subgroup trivial group The normal cores of the maximal subgroups of orders $6$ and $8$ intersect trivially.
Socle normal Klein four-subgroup of symmetric group:S4 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 $GL(2,3)$ whose inner automorphism group is $S_4$ (specifically $S_4$ is the quotient of $GL(2,3)$ 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 15 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.

### Short descriptions

Description Functions used Mathematical comment
SymmetricGroup(4) SymmetricGroup symmetric group of degree four (acting by default on $\{ 1,2,3,4 \}$. We can also specify a set on which we make the group act, e.g., SymmetricGroup([1,2,4,8])
PGL(2,3) PGL projective general linear group of degree two over the field of three elements