Alternating group:A4

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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

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 $PSL(2,3)$

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.

Endomorphisms

Automorphisms

The automorphism group of the alternating group of degree four is isomorphic to the symmetric group of degree four. Since the alternating group of degree four is centerless, it embeds as a subgroup inside its automorphism group.

Another way of thinking of this is as follows: in the symmetric group of degree four, the alternating group of degree four is a subgroup of index two, and every automorphism of this subgroup is realized as the restriction to this subgroup of an inner automorphism of the symmetric group.

Endomorphisms

The endomorphisms of the alternating group of degree four are given by the following (i.e., equivalent to one of these up to composition with an automorphism):

• The trivial map.
• The identity map.
• The retraction to a subgroup of order three, with kernel being the Klein four-group comprising the identity and the double transpositions. (All such retractions are equivalent).

Elements

Upto conjugacy

Further information: Splitting criterion for conjugacy classes in the alternating group

The alternating group on $\{1,2,3,4\}$ has four conjugacy classes. Two of these arise from other partitions of $4$ with an even number of cycles of even length, and with either a repetition of length or a cycle of even length. Two of these arise from a partition of $4$ into cycles of distinct odd length.

1. $4=1+1+1+1$, the identity element. (1)
2. $4=2+2$, the three double transpositions: $(1,2)(3,4),(1,3)(2,4),(1,4)(2,3)$. (3)
3. $4=3+1$, four of the $3$-cycles: $(1,2,3),(4,3,2),(3,4,1),(2,1,4)$. (4)
4. $4=3+1$, the remaining four $3$-cycles: $(1,3,2),(4,2,3),(3,1,4),(2,4,1)$. (4)

Upto automorphism

The conjugacy classes (1) and (2) are invariant under all automorphisms.

An outer automorphism interchanges classes (3) and (4). This can be realized, for instance, by viewing the alternating group as a subgroup of the symmetric group of degree four. Any transposition or $4$-cycle in the symmetric group interchanges classes (3) and (4).

Subgroups

Further information: Subgroup structure of alternating group:A4

The alternating group on $\{1,2,3,4\}$ has the following subgroups (clubbed together by conjugacy):

1. The trivial subgroup. (1)
2. Three subgroups of order two, each generated by a double transposition, such as $(1,2)(3,4)$. These are all isomorphic to the cyclic group of order two. (3)
3. A subgroup of order four, comprising the identity element and the three double transpositions: $\{(),(1,2)(3,4),(1,3)(2,4),(1,4)(2,3)\}$. These are all isomorphic to the Klein four-group. (1)
4. Four subgroups of order three, each generated by a $3$-cycle, such as $(1,2,3)$. These are all isomorphic to the cyclic group of order three. (4)
5. The whole group. (1)

There is no subgroup of order $6$. This is the smallest possible order of a group not having subgroups of all orders dividing the group order.

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 (this is type (3) in the list above).

Characteristic subgroups

In this group, the characteristic subgroups are the same as the normal subgroups. In other words, this is a group in which every normal subgroup is characteristic

Apart from the whole group and the trivial subgroup, there are four retracts -- the four Sylow 3-subgroups (listed as type (4) above). These all occur as retracts with the kernel being the subgroup formed by the double transpositions.

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(3);
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)