nLab alternating group

Idea

An alternating group $A_n$ is a subgroup of a symmetric group $S_n$ consisting of the even permutations.

The alternating group $A_n$ is to the symmetric group $S_n$ as the special orthogonal group $SO(n)$ is to the orthogonal group $O(n)$. See also at symmetric group – Whitehead tower

Examples

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

• The alternating group $A_5$ on five elements, of order $60$, 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 $n \geq 5$, the alternating group $A_n$ is simple. This is true even if $n$ is infinite: define $Alt(X)$ for any set $X$ to consist of all permutations of $X$ each of which fixes all but finitely elements, and which is an even permutation on that finite subset.

Related concepts

• finite group

• alternating representation

References

• Wikipedia, Alternating group