nLab
alternating group

Contents

Contents

Idea

An alternating group A nA_n is a subgroup of a symmetric group S nS_n consisting of the even permutations.

The alternating group A nA_n is to the symmetric group S nS_n as the special orthogonal group SO(n)SO(n) is to the orthogonal group O(n)O(n). See also at symmetric group – Whitehead tower

Examples

  • The alternating group A 4A_4 on four elements is isomorphic to the orientation-preserving tetrahedral group.

  • The alternating group A 5A_5 on five elements, of order 6060, is the smallest nonabelian simple group. Geometrically, it may be realized as finite subgroup of SO(3) which carries a regular icosahedron into itself: the icosahedral group.

  • For all n5n \geq 5, the alternating group A nA_n is simple. This is true even if nn is infinite: define Alt(X)Alt(X) for any set XX to consist of all permutations of XX each of which fixes all but finitely elements, and which is an even permutation on that finite subset.

References

Last revised on September 11, 2018 at 14:34:46. See the history of this page for a list of all contributions to it.