Alternating group:A4

Definition
The alternating group $$A_4$$ is defined in the following equivalent ways:


 * It is the group of even permutations (viz., the member of family::alternating group) on four elements.
 * It is the member of family::von Dyck group (sometimes termed triangle group, though triangle group has an alternative interpretation) with parameters $$(2,3,3)$$ (sometimes written in reverse order as $$(3,3,2)$$).
 * It is the group of orientation-preserving symmetries of a regular tetrahedron. When viewed in this light, it is called the tetrahedral group, and its symbol as a point group is $$T$$ or $$332$$.
 * It is the member of family::projective special linear group of degree two over the field of three elements, viz., $$PSL(2,3)$$.
 * It is the member of family::general affine group of degree $$1$$ over the field of four elements, viz., $$GA(1,4)$$ (also written as $$AGL(1,4)$$.

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).

Up to automorphism
The conjugacy classes of the identity element and double transpositions are invariant under all automorphisms.

An outer automorphism interchanges the conjugacy classes of elements of order three (each of size four). 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 those two conjugacy classes

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.

Other definitions
The alternating group can be constructed in many equivalent ways:


 * As the alternating group of degree four, using GAP's AlternatingGroup function:

AlternatingGroup(4)

F := FreeGroup(3); G := F/[F.1^3, F.2^3, F.3^2, F.1*F.2*F.3]
 * Using the von Dyck presentation. Here is a sequence of steps:

The output $$G$$ is the alternating group.
 * As the projective special linear group, using GAP's ProjectiveSpecialLinearGroup function:

PSL(2,3)