False statementConstruction: Literature-sourcedVerification: AI-generatedSession-authored (Fable 5 assisted)precheck passaudited 2026-08-27

How statement and proof provenance work

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.

FALSE: the socle is always a single simple group

Statement

False claim: for every finite group $G$ , the socle $soc (G)$ is a single simple subgroup.

Facts & Assumptions

Given: A nonabelian finite simple group $T$ and the direct product $G = T \times T$ .

[L1]

Finite characteristically simple groups are direct products of isomorphic simple groups (Finite characteristically simple groups are direct products of isomorphic simple groups).

[L2]

The socle of a finite group is a direct product of minimal normal subgroups (The socle is characteristic and decomposes as a direct product of minimal normal subgroups).

Refutation

technique · direct

1.1

In the group $G = T \times T$ , each factor $T \times 1$ and $1 \times T$ is a minimal normal subgroup, and they are distinct.

given

2.1

By [L2], the socle of $G$ is the direct product of those minimal normal subgroups, so $soc (G) = T \times T$ .

L2step 1.1

3.1

The group $T \times T$ is not simple because each factor is a proper nontrivial normal subgroup. Therefore the claim is false.

L1step 2.1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

9 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

James E. Humphreys, A Course in Group Theory, Proposition 16.11 (standard reference, not scraped)