ExampleConstruction: AI-adaptedVerification: AI-adaptedprecheck passaudited 2026-08-16

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For $n \geq 2$ , $S_{n} ≅ A_{n} ⋊ C_{2}$ using any transposition complement

Example

For every $n \geq 2$ and every transposition $τ \in S_{n}$ ,

$S_{n} ≅ A_{n} ⋊ ⟨ τ ⟩ ≅ A_{n} ⋊ C_{2} .$

Facts & Assumptions

Given: An integer $n \geq 2$ and a transposition $τ \in S_{n}$ .

[L1]

The sign map $sgn : S_{n} \to {1, - 1}$ is a surjective homomorphism for $n \geq 2$ (The sign is a homomorphism $S_{n} \to {+ 1, - 1}$ , surjective exactly when $n \geq 2$ ).

[L2]

The alternating group $A_{n}$ is the kernel of the sign homomorphism (The alternating group $A_{n} = \ker (sgn)$ of even permutations).

[L3]

The kernel of a group homomorphism is normal (The image of a group homomorphism is a subgroup and its kernel is a normal subgroup).

[L4]

A normal factor and a complement with trivial intersection give an external semidirect product by conjugation ( Recognition theorem: $G = N H$ with $N ⊴ G$ , $N \cap H = 1$ exactly realises an external semidirect product).

[L5]

$S_{n}$ consists of the permutations of an $n$ -element set (The symmetric group $Sym (X)$ : the bijections of a set $X$ under composition).

Verification

technique · direct

1.1L1L2L3L5

By [L1]--[L3], $A_{n}$ is normal. The transposition $τ$ has sign $- 1$ , so $⟨ τ ⟩ = {1, τ}$ intersects $A_{n}$ trivially.

1.2L1algebra

If $σ \in S_{n}$ is even, then $σ \in A_{n}$ . If it is odd, then $σ τ$ is even and $σ = (σ τ) τ$ . Hence $S_{n} = A_{n} ⟨ τ ⟩$ .

2.1step 1.1step 1.2L4∎

The recognition theorem [L4] gives the asserted decomposition, with the action on $A_{n}$ given by conjugation by $τ$ .

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

22 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

J. S. Milne, Group Theory (standard reference, not scraped)

For n≥2n≥2, Sn≅An⋊C2Sn​≅An​⋊C2​ using any transposition complement