# Symmetric group:S5

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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

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 .

The Jordan-Holder decomposition of is unique.

## Elements

### Upto conjugacy

For convenience, we take the underlying set to be .

There are seven conjugacy classes, corresponding to the unordered integer partitions of (for more information, refer cycle type determines conjugacy class). We use the notation of the cycle decomposition for permutations:

- , i.e., five fixed points: The identity element. (1)
- , i.e., one -cycle and three fixed points: The transpositions, such as . (10)
- , i.e., one -cycle and two fixed points: The -cycles, such as . (20)
- , i.e., one -cycle and one fixed point: The -cycles, such as . (30)
- , i.e., one -cycle: The -cycles, such as . (24)
- : Permutations such as . (20)
- : The double transpositions, such as . (15)

Of these, types (1),(3),(5),(7) are conjugacy classes of even permutations, while types (2), (4), and (6) are conjugacy classes of odd permutations. The even permutations together form a subgroup, namely, the alternating group of degree five.

### Upto automorphism

is a complete group: in particular, every automorphism of the group is inner. Thus, the equivalence classes under automorphisms are the same as the conjugacy classes.

## 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

## 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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