Symmetric group:S4
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Definition
Verbal definition
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: .
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 | (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 |
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
Lists of numerical invariants
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 | , 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 |
Endomorphisms
Automorphisms
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
Endomorphisms
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
Up to conjugacy
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 .
Up to automorphism
Since 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
Here is a complete list of subgroups. Note that since 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 . Isomorphic to cyclic group of order two.(6)
- The two-element subgroup generated by a double transposition, such as . Isomorphic to cyclic group of order two. (3)
- The four-element subgroup generated by two disjoint transpositions, such as . 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 are: the trivial subgroup (type (1)), the Klein four-group (type (5) -- note that type (4) are non-normal subgroups), the alternating group (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 -Sylow subgroups are dihedral groups of order eight, type (7) in the list.
- The -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 and 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 is contained in higher symmetric groups, most notably the symmetric group on five elements .
Extensions
These include whose inner automorphism group is (specifically is the quotient of 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 :
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:
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 . 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 |