ExampleConstruction: Literature-sourcedVerification: AI-adaptedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-04

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A_4 has four complements to its normal Klein four subgroup

Example

In $A_{4}$ , the normal Klein four subgroup

$V = {(), (12) (34), (13) (24), (14) (23)}$

has exactly four complements, namely the four subgroups generated by a $3$ -cycle.

Facts & Assumptions

Given: The alternating group $A_{4}$ (The alternating group $A_{n} = \ker (sgn)$ of even permutations) and its normal Klein four subgroup $V$ .

[L1]

A normal Hall subgroup has a complement (Schur-Zassenhaus existence theorem).

[L2]

Under the solvability hypothesis, any two complements are conjugate (Schur-Zassenhaus conjugacy when the kernel or quotient is solvable).

Verification

technique · direct

1.1

The subgroup $V$ has order $4$ and index $3$ , so it is a normal Hall subgroup of $A_{4}$ . The four subgroups $⟨ (123) ⟩$ , $⟨ (124) ⟩$ , $⟨ (134) ⟩$ , and $⟨ (234) ⟩$ each have order $3$ , intersect $V$ trivially, and together with $V$ generate $A_{4}$ , so they are complements.

givenL1algebra

2.1

There are exactly four subgroups of order $3$ in $A_{4}$ , because the eight $3$ -cycles come in inverse pairs and each order- $3$ subgroup has exactly two nonidentity elements. Hence the four displayed complements are all the complements to $V$ . Since $A_{4} / V ≅ C_{3}$ is solvable, [L2] also predicts that they form one conjugacy class.

L2step 1.1algebra∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

17 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

David A. Craven, Finite Group Theory (standard reference, not scraped)