ExampleConstruction: AI-adaptedVerification: AI-adaptedSession-authored (Fable 5 assisted)precheck passjudge pass (deepseek-v4-pro + gpt-5.6-terra)audited 2026-08-13

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Composition and derived series of $S_{4}$

Example

Let $V_{4} = {1, (12) (34), (13) (24), (14) (23)} \leq A_{4}$ and $K = ⟨ (12) (34) ⟩$ . Then $S_{4} ▹ A_{4} ▹ V_{4} ▹ K ▹ 1$ is a composition series with factor orders $2, 3, 2, 2$ , while the derived series is $S_{4} ▹ A_{4} ▹ V_{4} ▹ 1$ .

Facts & Assumptions

Given: The displayed subgroups of $S_{4}$ .

[F1]

A composition series is a strict subnormal chain with simple factors (Composition series, composition factors, and composition length).

[F2]

Derived length is the least index at which the derived series is trivial (The derived series, solvable groups, and derived length).

[L2]

$S_{4}^{'} = A_{4}$ ( $[S_{n}, S_{n}] = A_{n}$ for $n \geq 2$ , and $[A_{n}, A_{n}] = A_{n}$ for $n \geq 5$ ).

[L3]

For $N ⊴ G$ , the quotient $G / N$ is abelian if and only if $G^{'} \leq N$ ( $G / N$ is abelian if and only if $[G, G] \subseteq N$ ).

[L4]

The derived subgroup of a group is characteristic and hence normal (The derived subgroup is characteristic and the abelianization is universal).

Verification

technique · direct

1.1

The displayed terms have orders $24, 12, 4, 2, 1$ . Each is normal in the preceding term: $A_{4}$ is the sign kernel, $V_{4}$ is normal in $A_{4}$ , and $K$ is normal in the abelian group $V_{4}$ .

givenalgebra

1.2

By [L2], $S_{4}^{'} = A_{4}$ . The quotient $A_{4} / V_{4}$ has order three and is abelian, so [L3] gives $A_{4}^{'} \leq V_{4}$ . Direct calculation gives $[(123), (124)] = (12) (34)$ ; normality of $A_{4}^{'}$ from [L4] and conjugation by $A_{4}$ then put all three nonidentity elements of $V_{4}$ in $A_{4}^{'}$ . Thus $A_{4}^{'} = V_{4}$ , while $V_{4}^{'} = 1$ because $V_{4}$ is abelian. The derived series is therefore the displayed chain of length three by [F2].

L2L3L4F2algebra

2.1

The adjacent quotient orders are $2, 3, 2, 2$ , so the quotients are simple and the chain is a composition series by [F1].

step 1.1F1algebra

3.1

Thus the composition length is four while the derived length is three; the two series measure different features of $S_{4}$ .

step 2.1step 1.2∎

Depends on

Used by

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Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 53 results over 16 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click the chart to enlarge it.

Sources

J. S. Milne, Group Theory, Chapter 6 (standard reference, not scraped)
K. Conrad, Subgroup Series I (standard reference, not scraped)
K. Igusa, Notes on Jordan-Hölder, section 5 (standard reference, not scraped)

Composition and derived series of S4S4​