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$A_{n}$ is generated by $3$ -cycles for every $n \geq 3$

Statement

For every $n \geq 3$ , the alternating group $A_{n}$ is generated by its $3$ -cycles.

Facts & Assumptions

Given: $n \geq 3$ and the alternating group $A_{n}$ .

[F1]

$A_{n}$ is the kernel of the sign homomorphism (The alternating group $A_{n} = \ker (sgn)$ of even permutations, The sign is a homomorphism $S_{n} \to {+ 1, - 1}$ , surjective exactly when $n \geq 2$ ).

[F2]

Every permutation is a product of transpositions (Every finite permutation is a product of transpositions, so the transpositions generate $S_{n}$ ).

[F3]

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).

[F4]

The subgroup generated by a set is the smallest subgroup containing it (The subgroup $⟨ S ⟩$ generated by a subset, the cyclic subgroup $⟨ g ⟩$ , and cyclic groups).

Proof

technique · direct

1.1

Let $σ \in A_{n}$ . By [F2], write it as a product of transpositions; [F1] and [F3] imply that the number of factors is even.

F1F2F3

1.2

Pair consecutive transpositions. An equal pair cancels; two sharing one entry multiply to a $3$ -cycle; and two disjoint pairs satisfy $(a b) (c d) = (a c b) (a c d)$ .

algebra

2.1

Thus every $σ \in A_{n}$ is a product of $3$ -cycles.

step 1.1step 1.2

3.1

Conversely, [F3] makes every $3$ -cycle even, hence an element of $A_{n}$ by [F1]. Therefore [F4] and step 2.1 prove that the $3$ -cycles generate $A_{n}$ .

F1F3F4step 2.1∎

Depends on

Used by

Dependency tree · two levels

16 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)

AnAn​ is generated by 33-cycles for every n≥3n≥3