Alternating Groups

Introduction

The alternating groups are a critical family of simple groups. They are fundamental to understanding the structure of symmetric groups and have profound implications for Galois theory.

Definition

Definition 4.2:

A permutation is even if it can be written as a product of an even number of transpositions (2-cycles). It is odd otherwise.
The sign of a permutation $σ$ , denoted $sgn (σ)$ , is $+ 1$ if $σ$ is even and $- 1$ if $σ$ is odd.
The map $sgn : S_{n} \to {1, - 1}$ is a group homomorphism.
The alternating group $A_{n}$ is the kernel of this homomorphism.

Properties

As the kernel of a homomorphism, $A_{n}$ is a normal subgroup of $S_{n}$ . Its index is $[S_{n} : A_{n}] = 2$ , so its order is $| A_{n} | = n! / 2$ .

Structural Properties

The structural properties of alternating groups are fundamental to the entire theory:

Abelian Property

$A_{n}$ is abelian if and only if $n \leq 3$ .

Simplicity of $A_{4}$

$A_{4}$ is not simple. It contains a normal subgroup of order 4, the Klein four-group:

V = {e, (1, 2) (3, 4), (1, 3) (2, 4), (1, 4) (2, 3)}

Simplicity for $n \geq 5$

Theorem 4.3: The alternating group $A_{n}$ is simple for all $n \geq 5$ .

Examples

Example 1: $A_{3}$

The alternating group $A_{3}$ has order $3! / 2 = 3$ and consists of:

$e$ (identity)
$(1, 2, 3)$ (3-cycle)
$(1, 3, 2)$ (3-cycle)

$A_{3}$ is abelian and isomorphic to $Z_{3}$ .

Example 2: $A_{4}$

The alternating group $A_{4}$ has order $4! / 2 = 12$ and consists of:

The identity
All 3-cycles: $(1, 2, 3)$ , $(1, 2, 4)$ , $(1, 3, 2)$ , $(1, 3, 4)$ , $(1, 4, 2)$ , $(1, 4, 3)$ , $(2, 3, 4)$ , $(2, 4, 3)$
Products of two disjoint transpositions: $(1, 2) (3, 4)$ , $(1, 3) (2, 4)$ , $(1, 4) (2, 3)$

$A_{4}$ is not simple because it contains the Klein four-group as a normal subgroup.

Example 3: $A_{5}$

The alternating group $A_{5}$ has order $5! / 2 = 60$ and is simple. It contains:

The identity
All 3-cycles (20 elements)
All 5-cycles (24 elements)
Products of two disjoint transpositions (15 elements)

Historical Significance

The simplicity of $A_{n}$ for $n \geq 5$ is one of the most important results in elementary group theory. It is the deep structural reason behind the central result of Galois theory: the insolvability of the general polynomial equation of degree five or higher.

A group is called "solvable" if its composition factors are all abelian (specifically, cyclic of prime order). Since $A_{5}$ is a non-abelian simple group, any group having $A_{5}$ as a composition factor cannot be solvable.

The Galois group of a general quintic polynomial is $S_{5}$ , which has the composition series:

{e} ⊴ A_{5} ⊴ S_{5}

The factors are:

$A_{5} / {e} ≅ A_{5}$ (not abelian)
$S_{5} / A_{5} ≅ Z_{2}$ (abelian)

Because the factor $A_{5}$ is not abelian, the group $S_{5}$ is not solvable, which ultimately proves the impossibility of a general quintic formula involving only radicals.

Applications

Application 1: Galois Theory

The simplicity of $A_{n}$ for $n \geq 5$ is crucial for understanding the insolvability of polynomial equations of degree 5 or higher.

Application 2: Group Classification

Alternating groups are fundamental examples of simple groups and are essential to the classification of finite simple groups.

Application 3: Symmetry

Alternating groups represent the "even" symmetries of objects, which are often more fundamental than the full symmetric group.

Application 4: Combinatorics

Alternating groups are important in combinatorics, particularly in the study of permutations and their properties.