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# Alternating group:A4

### From Groupprops

*This article is about a particular group, viz a group unique upto isomorphism*

*This particular group is the smallest (in terms of order):* solvable non-nilpotent group

*This particular group is the smallest (in terms of order):* group not having subgroups of every order dividing the group order

*This particular group is a finite group of order:* 12

## Contents |

## Definition

The alternating group *A*_{4} is defined in the following equivalent ways:

- It is the group of even permutations (viz, the alternating group) on four elements
- It is the von Dyck group (sometimes termed
*triangle group*) with parameters (3,3,2) - It is the group of orientation-preserving symmetries of a regular tetrahedron
- It is the projective special linear group of order 2 over the field of three elements, viz
*P**S**L*(2,3)

## Group properties

### Solvability

*This particular group is solvable*

The commutator subgroup of *A*_{4} is the Klein group of order 4, namely the normal subgroup comprising double transpositions. This is Abelian.

Thus, *A*_{4} is solvable of solvable length 2, or in other words, it is a metabelian group.

### Nilpotence

*This particular group is not nilpotent*

### Abelianness

*This particular group is not Abelian*

### Simplicity

*This particular group is not simple*

Since *A*_{4} has a proper nontrivial commutator subgroup, it is not simple.

## Subgroups

### Normal subgroups

Apart from the trivial subgroup and the whole group, there is exactly one normal subgroup, namely the subgroup of order 4 comprising the identity element and the three double transpositions.

### Characteristic subgroups

*In this group, the characteristic subgroups are the same as the normal subgroups. In other words, this is a N=C-group*Template:Retracts

Apart from the whole group and the trivial subgroup, there are four retracts -- the four Sylow 3-subgroups (which are all conjugate to each other). These all occur as retracts with the kernel being the subgroup formed by the double transpositions.

## Endomorphisms

The automorphisms of the alternating group form the symmetric group *S*_{4}. A convenient way of thinking of this is by embedding the alternating group *A*_{4} inside the symmetric group *S*_{4}, and then observing that since it is a normal subgroup, the symmetric group acts on the alternating group by conjugation. This gives a homomorphism from *S*_{4} to *A**u**t*(*A*_{4}). A bit of checking shows that this map is an isomorphism.

The analogous statement is true for most alternating groups.

## Supergroups

*These are groups containing the alternating group*

The alternating group is contained in the symmetric group on 4 elements, as a normal subgroup of index two. It is, in fact, a fully characteristic subgroup. The complement exists as a subgroup, namely that generated by a transposition.

## Subgroup-defining functions

### Center

*The center of this group is abstractly isomorphic to*: trivial group

The alternating group is a centerless group, viz its center is the trivial subgroup.

### Commutator subgroup

*The commutator subgroup of this group is abstractly isomorphic to:* Klein-four group

The commutator subgroup of the alternating group is the Klein-four group, comprising the identity element, and the three double transpositions.

## Quotient-defining functions

### Inner automorphism group

*The inner automorphism group of this group, viz the quotient group by its center, is abstractly isomorphic to:* the whole group

### Abelianization

*The Abelianization of this group, viz the quotient group by its commutator subgroup, is abstractly isomorphic to* cyclic group of order three

## Extensions

*These are groups having the alternating group as a quotient group*
Perhaps the most important of these is *S**L*(2,3), which is the universal central extension of *P**S**L*(2,3). The kernel of the projection mapping is a two-element subgroup, namely the identity matrix and the negative identity matrix.

## Implementation using GAP

### Group ID

The alternating group is the *third* group of order 12 in the small-group enumeration using GAP. Thus, it can be defined in GAP as:

SmallGroup(12,3)

### Other definitions

The alternating group can be constructed in many equivalent ways:

- As the alternating group. The command is
AlternatingGroup(4)

- Using the von Dyck presentation. Here is a sequence of steps:

F := FreeGroup["g","h","k"]; G := F/[F.1^3, F.2^3, F.3^2, F.1*F.2*F.3]

The output *G* is the alternating group.

- As the projective special linear group. The command is
PSL(2,3)