# Symmetric group:S4

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## Definition

### Permutation definition

The symmetric group is defined as the group of all permutations on a set of 4 elements.

### Presentation

## Group properties

### Solvability

*This particular group is solvable*

The commutator subgroup of is , whose commutator subgroup is the Klein group of order 4, namely the normal subgroup comprising double transpositions. This is Abelian.

Thus, 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 has a nontrivial commutator subgroup, it is not Abelian.

There exist subnormal subgroups of which are not hypernormalized. For instance, the subgroup generated by the double transposition 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 (a dihedral group) which is not normal.

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

## Endomorphisms

### Automorphisms

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

### 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 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

All the four endomorphisms described above are retractions. Thus, in fact, every endomorphism of is equivalent, via some automorphism, to a retraction.

## Subgroups

### Normal subgroups

The only normal subgroups of are: the whole group, the trivial subgroup, , and the Klein-four group.

### Characteristic subgroups

Since every automorphism of is inner, the characteristic subgroups of are precisely the same as the normal subgroups.

### Fully characteristic subgroups

In fact, all the normal subgroups described above are fully characteristic, a fact that can easily be checked from the explicit description of endomorphisms.

### Retraction-invariant subgroups

Again, the retraction-invariant subgroups are precisely the same as the normal subgroups, as can be shown from the explicit description of retractions given above.

### 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 are precisely the same as the normal subgroups.

### Conjugate-permutable subgroups

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.

### Hypernormalized subgroups

These are precisely the same as the normal subgroups.