Alternating group
From Groupprops
Contents
Definition
For a finite set
Let be a finite set. The alternating group on
is defined in the following equivalent ways:
- It is the group of all even permutations on
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
is the subgroup of the symmetric group on
comprising the even permutations.
- It is the kernel of the sign homomorphism from the symmetric group on
to the group
.
For having size zero or one, the alternating group on
equals the whole symmetric group on
. For
having size at least two, the alternating group on
is the unique subgroup of index two in the symmetric group on
.
The alternating group on a set of size is denoted
and is termed the alternating group of degree
.
For an infinite set
Let be an infinite set. The finitary alternating group on
is defined in the following equivalent ways:
- It is the group of all even permutations on
under composition.
- It is the kernel of the sign homomorphism on the finitary symmetric group on
.
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, is the degree of the alternating group, i.e., the size of the set it acts on.
For all the statements involving , we use the fact that A5 is simple and alternating groups are simple for degree at least five.
Function | Value | Explanation |
---|---|---|
order | ![]() ![]() ![]() ![]() |
It has index two in the symmetric group of degree ![]() |
exponent | ![]() ![]() ![]() ![]() |
|
nilpotency class | ![]() ![]() ![]() |
abelian group for ![]() ![]() |
derived length | ![]() ![]() ![]() ![]() ![]() |
abelian for ![]() ![]() |
Frattini length | ![]() ![]() ![]() ![]() |
Frattini-free group: intersection of maximal subgroups is trivial. |
Fitting length | ![]() ![]() ![]() ![]() ![]() |
abelian for ![]() ![]() |
minimum size of generating set | 2 | |
subgroup rank | At most ![]() |
|
max-length | ? | |
chief length | ![]() ![]() ![]() ![]() ![]() ![]() |
|
composition length | ![]() ![]() ![]() ![]() ![]() ![]() |
Group properties
Property | Satisfied | Explanation |
---|---|---|
Abelian group | Yes for ![]() ![]() |
![]() ![]() |
Nilpotent group | Yes for ![]() ![]() |
![]() |
Solvable group | Yes for ![]() ![]() |
alternating groups are simple for degree five or more. |
Supersolvable group | Yes for ![]() ![]() |
|
Simple group | Yes for ![]() ![]() ![]() |
|
Rational group | No for all ![]() |
|
Ambivalent group | Yes for ![]() |
See classification of ambivalent alternating groups. |