TheoremStatement: Literature-sourcedProof: AI-adaptedprecheck passjudge pass (deepseek-v4-pro + gpt-5.6-terra)audited 2026-08-13

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The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

$A_{5}$ and $S_{n}$ for $n \geq 5$ are not solvable

Statement

The alternating group $A_{n}$ is not solvable for every $n \geq 5$ . Consequently $S_{n}$ is not solvable for every $n \geq 5$ ; in particular, $A_{5}$ and $S_{5}$ are not solvable.

Facts & Assumptions

Given: An integer $n \geq 5$ .

[F1]

A group is solvable exactly when its derived series reaches the trivial group (The derived series, solvable groups, and derived length).

[L1]

Every subgroup of a solvable group is solvable (Subgroups and quotients of solvable groups are solvable).

[L2]

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

[L3]

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

Proof

technique · direct

1.1

By [L3], every positive term of the derived series of $A_{n}$ equals $A_{n}$ , which is nontrivial by [L2]; hence the series never reaches $1$ , and $A_{n}$ is not solvable by [F1].

givenL2L3F1

2.1

Since $A_{n} \leq S_{n}$ , solvability of $S_{n}$ would imply solvability of $A_{n}$ by [L1], contradicting step 1.1.

step 1.1L1

3.1

Therefore $A_{n}$ and $S_{n}$ are nonsolvable for all $n \geq 5$ , including $n = 5$ .

step 1.1step 2.1∎

Depends on

Used by

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

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)

A5A5​ and SnSn​ for n≥5n≥5 are not solvable