# Symmetric group:S5

This article is about a particular group, i.e., a group unique upto isomorphism. View specific information (such as linear representation theory, subgroup structure) about this groupView a complete list of particular groups (this is a very huge list!)[SHOW MORE]

## Definition

### Permutation definition

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

### Presentation

## Group properties

The commutator subgroup of is , 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 has a proper nontrivial commutator subgroup, it is not simple.

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 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 are all retractions.

## Subgroups

### Normal subgroups

The only normal subgroups of are: the whole group, the trivial subgroup, and (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 is a T-group (that is, normality is transitive).

This follows from the fact that 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 .

### Abnormal subgroups

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