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# Symmetric group:S4

### From Groupprops

*This article is about a particular group, viz a group unique upto isomorphism*

*This particular group is a finite group of order:* 24

## Contents |

## Definition

The symmetric group *S*_{4} is defined in the following equivalent ways:

- The group of all permutations, viz the symmetric group on a set of 4 elements
- The triangle group (
*not*the von Dyck group, but its double) Δ(3,3,2) - The group of all (not necessarily orientation-preserving) symmetries of the regular tetrahedron
- The group of orientation-preserving symmetries of the cube (or equivalently, the octahedron)
- The projective general linear group of order two over the field of three elements, viz:
*P**G**L*(2,3)

### Presentation

## Group properties

### Solvability

*This particular group is solvable*

The commutator subgroup of *S*_{4} is *A*_{4}, whose commutator subgroup is the Klein group of order 4, namely the normal subgroup comprising double transpositions. This is Abelian.

Thus, *S*_{4} is solvable of solvable length 3.

### Nilpotence

*This particular group is not nilpotent*

### Abelianness

*This particular group is not Abelian*

### Simplicity

*This particular group is not simple*

Since *S*_{4} has a proper onntrivial commutator subgroup, it is not simple.

There exist subnormal subgroups of *S*_{4} which are not hypernormalized. For instance, the subgroup generated by the double transposition (12)(34) is2-subnormal (because it is normal in the subgroup generated by all double transpositions, which in turn is normal). However, it is not a hypernormalized subgroup, because its normalizer is a group of order 8 (a dihedral group) which is not normal.

*S*_{4} is a centerless group, and moreover, every automorphism of *S*_{4} is inner. This can easily be checked by studying the
effect of any automorphism oon the set of transpositions in *S*_{4}.

## 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 endomorphism to a group of order two, given by the sign homomorphism
- The endomorphism to the symmetric group on 3 elements, with kernel being the Klein four-group

### Retractions

The endomorphisms of *S*_{4} are all retractions.

## Subgroups

### Normal subgroups

The only normal subgroups of *S*_{4} are: the whole group, the trivial subgroup, *A*_{4}, and the Klein-four group.

### Characteristic subgroups

*In this group, the characteristic subgroups are the same as the normal subgroups. In other words, this is a N=C-group*

This is because every automorphism of *S*_{4} is inner.

### Fully characteristic subgroups

*In this group, the fully characteristic subgroups are the same as the normal subgroups. In other words, this is a N=FC-group*

This can easily be checked from the explicit description of endomorphisms.

### Retraction-invariant subgroups

*The retraction-invariant subgroups of this group are the same as the normal subgroups*

This can easily be checked from the explicit description of retractions.

### Subnormal subgroups

Apart from the normal subgroups, the only subnormal subgroups are the two-element subgroups corresponding to the double transpositions.

### Permutable subgroups

*The permutable subgroups of this group are same as the normal subgroups. In other words, this is a N=P-group*

The permutable subgroups are precisely the same as the normal subgroups.

### Conjugate-permutable subgroups

*The conjugate-permutable subgroups of this group are same as the subnormal subgroups. In other words, this is a CPT-group*

The conjugate-permutable subgroups are precisely the same as the subnormal subgroups. In other words, apart from the normal subgroups, the two-element subgroups generated by double transpositions are also conjugate-permutable.

Template:Hypernormalized subgroups same as normal

Template:Transitively normal subgroups

The only transitively normal subgroups of *S*_{4} are the whole group, the trivial subgroup, and the alternating group. The Klein-four group is not transitively normal.

The only central factors are the whole group and the trivial subgroup.

### Abnormal subgroups

The two proper abnormal subgroups (upto conjugacy) are: the symmetric group on three elemenst (which is of order 6), and the dihedral group on four elements (which is of order 8). These both occur as normalizers of Sylow subgroups (the one of order 8 is itself a Sylow subgroup, the other is the normalizer of a 3-cycle).

### Contranormal subgroups

Apart from the two abnormal subgroups above, the two-element subgroup comprising a single transposition, is also a contranormal subgroup. So is the four-element subgroup comprising a single 4-cycle.

Template:Self-normalizing subgroups same as abnormal

The subgroups that occur as retracts are: the whole group, the trivial subgroup, any two-element group for a single transposition, and any conjugate of the symmetric group on three elements.

## 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).

## Arithmetic functions

### Conjugacy classes

The symmetric group has five conjugacy classes, of which three are even and two are odd. The odd ones are the 4-cycle and the transposition. The even ones are the 3-cycle, the double transposition, and the identity element.

## Implementation in GAP

### Group ID

The symmetric group on 4 elements is the 12th group of order 24, in GAP's small-group enumeration. So it can be described as:

SmallGroup(24,12)