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# Alternating group:A5

### From Groupprops

*This article is about a particular group, viz a group unique upto isomorphism*

*This particular group is the smallest (in terms of order):* simple non-Abelian group

*This particular group is the smallest (in terms of order):* non-solvable group

*This particular group is the smallest (in terms of order):* nontrivial perfect group

## Contents |

## Definition

The alternating group *A*_{5} is defined in the following ways:

- It is the group of even permutations (viz, the alternating group) on five elements
- It is the von Dyck group (sometimes termed triangle group) with parameters (5,3,2)
- It is the group of orientation-preserving symmetries of the regular icosahedron (or equivalently, regular dodecahedron)
- It is the projective special linear group of order two over the field of five elements, viz
*P**S**L*(2,5)

## Group properties

## Subgroups

## Endomorphisms

## Bigger groups

### Groups having it as a subgroup

The alternating group is a subgroup of index two inside the symmetric group on five elements. It is also of index two in the full icosahedral symmetry group, which turns out *not* to be *S*_{5}, but instead the direct product of *A*_{5} and the cyclic group of order two.

### Groups having it as a quotient

The alternating group is a quotient of *S**L*(2,5) by its center. Hence, it is the inner automorphism group of *S**L*(2,5). *S**L*(2,5) is also the universal central extension of the alternating group.