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$A_{n}$ is simple for every $n \geq 5$

Statement

The alternating group $A_{n}$ is simple for every $n \geq 5$ .

Facts & Assumptions

Given: $n \geq 5$ .

[F1]

A group is simple when it is nontrivial and its only normal subgroups are the trivial subgroup and the whole group (Simple groups).

[F2]

Every nontrivial normal subgroup of $A_{n}$ contains a $3$ -cycle (Every nontrivial normal subgroup of $A_{n}$ contains a $3$ -cycle for $n \geq 5$ ).

[F3]

A normal subgroup of $A_{n}$ containing a $3$ -cycle is all of $A_{n}$ (A normal subgroup of $A_{n}$ containing one $3$ -cycle equals $A_{n}$ for $n \geq 5$ ).

[F4]

The group $A_{n}$ is the kernel of sign (The alternating group $A_{n} = \ker (sgn)$ of even permutations), and a $k$ -cycle has sign $(- 1)^{k - 1}$ (A $k$ -cycle has sign $(- 1)^{k - 1}$ , and $sgn (σ) = (- 1)^{n - c (σ)}$ when fixed points are counted as cycles).

Proof

technique · direct

1.1

The cycles $(1 2 3)$ and $(3 4 5)$ have sign $(- 1)^{2} = + 1$ by [F4], so they belong to $A_{n}$ ; they do not commute, so $A_{n}$ is nontrivial.

F4algebra

1.2

Let $N ⊴ A_{n}$ . If $N$ is nontrivial, [F2] gives a $3$ -cycle in $N$ , and [F3] then gives $N = A_{n}$ .

F2F3

2.1

Thus the only normal subgroups are ${1}$ and $A_{n}$ ; together with step 1.1, [F1] proves simplicity.

F1step 1.1step 1.2∎

Depends on

Used by

[Sₙ,Sₙ]=Aₙ for n≥2, and [Aₙ,Aₙ]=Aₙ for n≥5 Corollary
For n≥5, the only proper nontrivial normal subgroup of Sₙ is Aₙ Corollary
The Fitting subgroup of A₅ does not contain its centralizer Counterexample
A5 as the smallest nonabelian simple group Example
FALSE: Aₙ is simple for every n≥4 False statement
Cyclic and alternating simple families Remark
A₅ and Sₙ for n≥5 are not solvable Theorem

Dependency tree · two levels

15 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

T. Judson, Abstract Algebra: Theory and Applications, Simplicity of $A_n$ (standard reference, not scraped)

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