Alternating group

This article is about a basic definition in group theory. The article text may, however, contain advanced material.
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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 $S$.

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
• 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.

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	$\mathrm{l}\mathrm{c}\mathrm{m}\{1,2,\dots ,n-2,n\}$ (skipping over $n-1$) if $n$ is odd, $\mathrm{l}\mathrm{c}\mathrm{m}\{1,2,\dots ,n-1\}$ 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.