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Alternating group:A5

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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): 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 A5 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 PSL(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 S5, but instead the direct product of A5 and the cyclic group of order two.

Groups having it as a quotient

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


Implementation using GAP

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