# 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 $S_4$.

The Jordan-Holder decomposition of $S_5$ is unique.

## Elements

### Upto conjugacy

For convenience, we take the underlying set to be $\{ 1,2,3,4,5 \}$.

There are seven conjugacy classes, corresponding to the unordered integer partitions of $5$ (for more information, refer cycle type determines conjugacy class). We use the notation of the cycle decomposition for permutations:

1. $5 = 1 + 1 + 1 + 1 +1$, i.e., five fixed points: The identity element. (1)
2. $5 = 2 + 1 + 1 + 1$, i.e., one $2$-cycle and three fixed points: The transpositions, such as $(1,2)$. (10)
3. $5 = 3 + 1 + 1$, i.e., one $3$-cycle and two fixed points: The $3$-cycles, such as $(1,2,3)$. (20)
4. $5 = 4 + 1$, i.e., one $4$-cycle and one fixed point: The $4$-cycles, such as $(1,2,3,4)$. (30)
5. $5 = 5$, i.e., one $5$-cycle: The $5$-cycles, such as $(1,2,3,4,5)$. (24)
6. $5 = 3 + 2$: Permutations such as $(1,2,3)(4,5)$. (20)
7. $5 = 2 + 2 + 1$: The double transpositions, such as $(1,2)(3,4)$. (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

$S_5$ 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 $S_5$ 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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