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The symmetric group or , also termed the symmetric group of degree four, is defined in the following equivalent ways:
- The group of all permutations, i.e., the symmetric group 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) . In other words, it has the presentation:
where 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 .
- The von Dyck group with parameters . In other words, it has the presentation:
- The octahedral group or cube group: group of orientation-preserving symmetries of the cube (or equivalently, the octahedron). This is denoted as .
- The projective general linear group of degree two over the field of three elements: .
- The general affine group of degree two over the field of two elements: .
- The projective special linear group of degree two over ring:Z4, the ring of the integers modulo 4.
Equivalence of definitions
The following is a list of proofs of the equivalence of various definitions:
- Full tetrahedral group is isomorphic to S4
- von Dyck group with parameters (4,3,2) is isomorphic to S4
- PGL(2,3) is isomorphic to S4
- GA(2,2) is isomorphic to S4
- PSL(2,Z4) is isomorphic to S4
|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||(4,3,2)||click here for a list||is a spherical von Dyck group, i.e., it occurs as a finite subgroup of . In particular, this makes it a Coxeter group. Further information: Classification of finite subgroups of SO(3,R)|
|triangle group||(3,3,2)||click here for a list||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|
Want to compare and contrast arithmetic function values with other groups of the same order? Check out groups of order 24#Arithmetic functions
Lists of numerical invariants
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.
|Abelian group||No||, don't commute||is non-abelian, .|
|Nilpotent group||No||Centerless: The center is trivial||is non-nilpotent, .|
|Metacyclic group||No||No cyclic normal subgroup||is not metacyclic, .|
|Supersolvable group||No||No cyclic normal subgroup||is not supersolvable, .|
|Solvable group||Yes||Length three, commutator subgroup is , its commutator is Klein four-group||Largest for which is solvable.|
|T-group||No||Double transposition generates non-normal 2-subnormal subgroup||Only for which isn't a T-group.|
|HN-group||No||Double transposition generates subnormal non-hypernormalized subgroup||Only for which isn't hypernormalized.|
|Complete group||Yes||Centerless and every automorphism's inner||Symmetric groups are complete except the ones of degree .|
|Monolithic group||Yes||Monolith is the Klein four-group of double transpositions||All symmetric groups are monolithic; 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 .|
|Group having subgroups of all orders dividing the group order||Yes||(See subgroup list)||Largest 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|
Since is a complete group, it is isomorphic to its automorphism group, where each element of acts on by conjugation. In fact, for , the symmetric group is a complete group. Further information: Symmetric groups on finite sets are complete
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).
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 .
|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||even; no||1|
|2 + 1 + 1||2 (1 time), 1 (2 times)||one transposition (cycle of size two), two fixed points||, , , , ,||6||, also||odd||2|
|2 + 2||2 (2 times)||double transposition: two cycles of size two||, ,||3||even; no||2|
|3 + 1||3 (1 time), 1 (1 time)||one 3-cycle, one fixed point||, , , , , , ,||8||or||even; yes; no||3|
|4||4 (1 time)||one 4-cycle, no fixed points||, , , , ,||6||or||odd||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
Further information: Subgroup structure of symmetric group:S4
|Number of subgroups|| 30|
Compared with : 1,2,6,30,156,1455,11300, 151221
|Number of conjugacy classes of subgroups|| 11|
Compared with : 1,2,4,11,19,56,96,296,554,1593
|Number of automorphism classes of subgroups|| 11|
Compared with : 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.
|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 and 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.|
Linear representation theory
Further information: Linear representation theory of symmetric group:S4
|Degrees of irreducible representations over a splitting field|| 1,1,2,3,3 |
maximum: 3, lcm: 6, number: 5, sum of squares: 24, quasirandom degree: 1
|Schur index values of irreducible representations|| 1,1,1,1,1 |
maximum: 1, lcm: 1
|Smallest ring of realization for all irreducible representations (characteristic zero)|| |
Same as ring generated by character values
|Smallest field of realization for all irreducible representations, i.e., minimal splitting field (characteristic zero)|| (hence it is a rational representation group)|
Same as field generated by character values
|Condition for being a splitting field for this group||Any field of characteristic not two or three is a splitting field.|
|minimal splitting field in characteristic||prime field|
|Smallest size splitting field||Field:F5, i.e., the field with five elements.|
|Representation/Conjugacy class representative and size||(identity element) (size 1)||(size 3)||(size 6)||(size 6)||(size 8)|
|Irreducible representation of degree two with kernel of order four||2||2||0||0||-1|
|Product of standard and sign representations||3||-1||-1||1||0|
Further information: supergroups of symmetric group:S4
The symmetric group is contained in higher symmetric groups, most notably the symmetric group on five elements .
These include whose inner automorphism group is (specifically is the quotient of by its scalar matrices).
Implementation in GAP
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:
For instance, we can use the following assignment in GAP to create the group and name it :
gap> G := SmallGroup(24,12);
Conversely, to check whether a given group is in fact the group we want, we can use GAP's IdGroup function:
IdGroup(G) = [24,12]
or just do:
to have GAP output the group ID, that we can then compare to what we want.
|Description||Functions used||Mathematical comment|
|SymmetricGroup(4)||SymmetricGroup||symmetric group of degree four (acting by default on . 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|