# 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 $S$.

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 $S$.

## 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.

Function Value Explanation
order $n!/2$ for $n \ge 2$, $1$ for $n = 0,1$ It has index two in the symmetric group of degree $n$.
exponent $\operatorname{lcm}\{ 1,2,\dots,n \}$ if $n$ is odd, $\operatorname{lcm} \{ 1,2,\dots,n-1 \}$ if $n$ is even.
nilpotency class $1$ for $n \le 3$, undefined for $n \ge 4$ abelian group for $n \le 3$, nilpotent group for $n \ge 4$.
derived length $1$ for $n \le 3$, $2$ for $n = 4$, undefined for $n \ge 5$ abelian for $n \le 3$, simple for $n \ge 5$.
Frattini length $0$ for $n \le 2$, $1$ for $n \ge 3$ Frattini-free group: intersection of maximal subgroups is trivial.
Fitting length $1$ for $n \le 3$, $2$ for $n = 4$, undefined for $n \ge 5$ abelian for $n \le 3$, simple for $n \ge 5$.
minimum size of generating set 2
subgroup rank At most $n/2$
max-length  ?
chief length $0$ for $n \le 2$, $1$ for $n = 3, n \ge 5$, $2$ for $n = 4$.
composition length $0$ for $n \le 2$, $1$ for $n = 3, n \ge 5$, $3$ for $n = 4$.

## Group properties

Property Satisfied Explanation
Abelian group Yes for $n \le 3$, no for $n \ge 4$ $(1,2,3)$ and $(1,2)(3,4)$ don't commute.
Nilpotent group Yes for $n \le 3$, no for $n \ge 4$ $A_4$ is centerless.
Solvable group Yes for $n \le 4$, no for $n \ge 5$ alternating groups are simple for degree five or more.
Supersolvable group Yes for $n \le 3$, no for $n \ge 4$
Simple group Yes for $n = 3$ and $n \ge 5$, no for $n = 4$.
Rational group No for all $n$.
Ambivalent group Yes for $n = 1,2,5,6,10,14$, no otherwise See classification of ambivalent alternating groups.