Alternating group:A5

Definition
The alternating group $$A_5$$, also denoted $$\operatorname{Alt}(5)$$, and termed the alternating group of degree five, is defined in the following ways:


 * 1) It is the group of even permutations (viz., the member of family::alternating group) on five elements.
 * 2) It is the member of family::von Dyck group (sometimes termed triangle group, though the latter has a slightly different meaning) with parameters $$(2,3,5)$$ (sometimes written in reverse order as $$(5,3,2)$$).
 * 3) It is the icosahedral group, i.e., the group of orientation-preserving symmetries of the regular icosahedron (or equivalently, regular dodecahedron). Viewed this way, it is denoted $$l$$ or $$532$$.
 * 4) It is the member of family::projective special linear group of degree two over the field of four elements, viz., $$PSL(2,4)$$. It is also the member of family::special linear group of degree two over the field of four elements, i.e., $$SL(2,4)$$. It is also the member of family::projective general linear group of degree two over the field of four elements, i.e., $$PGL(2,4)$$.
 * 5) It is the member of family::projective special linear group of degree two over the field of five elements, viz., $$PSL(2,5)$$.

Equivalence of definitions

 * The equivalence of the definitions within (4) is given by isomorphism between linear groups when degree power map is bijective.
 * PGL(2,4) is isomorphic to A5: equivalence of (1) and (4)
 * PSL(2,5) is isomorphic to A5: equivalence of (1) and (5)

Automorphisms
The automorphism group of $$A_5$$ is the symmetric group on five letters $$S_5$$, with $$A_5$$ embedded in it as inner automorphisms.

Concretely, we can think of $$A_5$$ as embedded in $$S_5$$, and $$S_5$$ acting on $$A_5$$ by conjugation. The automorphisms obtained this way are all the automorphisms of $$A_5$$.

Other endomorphisms
Since $$A_5$$ is a finite simple group, it is a group in which every endomorphism is trivial or an automorphism. In particular, the endomorphisms of $$A_5$$ are: the trivial map, and the automorphisms described above.

Up to automorphism
Under outer automorphisms, the fourth and fifth conjugacy classes get merged. Thus, the classes under automorphism are of size $$1,15,20,24$$.

GAP implementation
Equivalent command: Running this command constructs the group as AlternatingGroup(5).

Memory usage: The memory usage for the SmallGroup construction is 1452.