# Symmetric group:S4

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## Definition

### Permutation definition

The symmetric group $S_4$ is defined as the group of all permutations on a set of 4 elements.

## 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 nontrivial commutator subgroup, it is not Abelian.

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

All the four endomorphisms described above are retractions. Thus, in fact, every endomorphism of $S_4$ is equivalent, via some automorphism, to a retraction.

## 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

Since every automorphism of $S_4$ is inner, the characteristic subgroups of $S_4$ 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.