Alternating group

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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

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-2,n \} (skipping over n - 1) 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.