Alternating group

Definition

For a finite set

Let $S$ be a finite set. The alternating group on $S$ is defined in the following equivalent ways:

1. It is the group of all even permutations on $S$ under composition. An even permutation is a permutation whose cycle decomposition has an even number of cycles of even size. Specifically, the alternating group on $S$ is the subgroup of the symmetric group on $S$ comprising the even permutations.
2. It is the kernel of the sign homomorphism from the symmetric group on $S$ to the group $\pm 1$.

For $S$ having size zero or one, the alternating group on $S$ equals the whole symmetric group on $S$. For $S$ having size at least two, the alternating group on $S$ is the unique subgroup of index two in the symmetric group on $S$.

For an infinite set

Let $S$ be an infinite set. The finitary alternating group on $S$ is defined in the following equivalent ways:

1. It is the group of all even permutations on $S$ under composition.
2. It is the kernel of the sign homomorphism on the finitary symmetric group on $S$.

Facts

• The alternating group on a set of size five or more is simple. Also, the finitary alternating group on an infinite set is simple. For full proof, refer: A5 is simple, alternating groups are simple