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Alternating group:A4

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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 A4 is defined in the following equivalent ways:

Group properties

Solvability

This particular group is solvable

The commutator subgroup of A4 is the Klein group of order 4, namely the normal subgroup comprising double transpositions. This is Abelian.

Thus, A4 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 A4 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-groupTemplate: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

Template:Automorphism group

The automorphisms of the alternating group form the symmetric group S4. A convenient way of thinking of this is by embedding the alternating group A4 inside the symmetric group S4, 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 S4 to Aut(A4). 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 SL(2,3), which is the universal central extension of PSL(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.

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