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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): group not having subgroups of every order dividing the group order
This particular group is a finite group of order: 12
The alternating group A4 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)
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.
This particular group is not nilpotent
This particular group is not Abelian
This particular group is not simple
Since A4 has a proper nontrivial commutator subgroup, it is not simple.
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.
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.
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.
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.
The alternating group is a centerless group, viz its center is the trivial subgroup.
The commutator subgroup of the alternating group is the Klein-four group, comprising the identity element, and the three double transpositions.
Inner automorphism group
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
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:
The alternating group can be constructed in many equivalent ways:
- As the alternating group. The command is
- 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