# Symmetric group:S5

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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

The commutator subgroup of $S_5$ is $A_5$, which is simple and hence not solvable.

### Nilpotence

This particular group is not nilpotent

### Abelianness

This particular group is not Abelian

### Simplicity

This particular group is not simple

Since $S_5$ has a proper nontrivial commutator subgroup, it is not simple. $S_5$ 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 $S_5$ is a complete group, it is isomorphic to its automorphism group, where each element of $S_4$ acts on $S_4$ by conjugation.

### Endomorphisms $S_5$ admits three kinds of endomorphisms (that is, it admits more endomorphisms, but any endomorphism is equivalent via an automorphism to one of these three):

• The endomorphism to the trivial group
• The identity map
• The endomorphism to a group of order two, given by the sign homomorphism

### Retractions

The endomorphisms of $S_5$ are all retractions.

## Subgroups

### Normal subgroups

The only normal subgroups of $S_5$ are: the whole group, the trivial subgroup, and $A_5$ (the alternating group).

### Characteristic subgroups

These are the same as the normal subgroups.

### Fully characteristic subgroups

These are the same as the normal subgroups.

### Retraction-invariant subgroups

These are the same as the normal subgroups.

### Subnormal subgroups

These are the same as the normal subgroups. Thus $S_5$ is a T-group (that is, normality is transitive).

This follows from the fact that $A_5$ is a simple group.

### Permutable subgroups

These are the same as the normal subgroups.

### Conjugate-permutable subgroups

These are the same as the normal subgroups.

### Contranormal subgroups

A subgroup is contranormal if and only if it is not contained within $A_5$.

### Abnormal subgroups

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