Alternating group

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 defining ingredient::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 defining ingredient::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 $$S$$.

The alternating group on a set of size $$n$$ is denoted $$A_n$$ and is termed the alternating group of degree $$n$$.

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.
 * Projective special linear group equals alternating group in only finitely many cases

Arithmetic functions
Here, $$n$$ is the degree of the alternating group, i.e., the size of the set it acts on.

For all the statements involving $$n \ge 5$$, we use the fact that A5 is simple and alternating groups are simple for degree at least five.