LemmaStatement: Literature-sourcedProof: AI-generatedSession-authored (Fable 5 assisted)precheck passjudge pass (gpt-5.6-terra)audited 2026-08-27

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Minimal normal subgroups of finite groups are characteristically simple

Statement

Every minimal normal subgroup of a finite group is characteristically simple.

Facts & Assumptions

Given: A finite group $G$ and a minimal normal subgroup $M ⊴ G$ .

[L1]

If $K$ is characteristic in a normal subgroup $M ⊴ G$ , then $K$ is normal in $G$ (If $K$ is characteristic in $N$ and $N$ is normal in $G$ , then $K$ is normal in $G$ ).

[A1]

A finite group is characteristically simple exactly when it has no proper nontrivial characteristic subgroup.

Proof

technique · direct

1.1

Let $K$ be a characteristic subgroup of $M$ . By [L1], the subgroup $K$ is normal in $G$ . Since $K \leq M$ and $M$ is minimal normal in $G$ , either $K = 1$ or $K = M$ .

givenL1

2.1

Thus $M$ has no proper nontrivial characteristic subgroup, so [A1] shows that $M$ is characteristically simple.

A1step 1.1∎

Depends on

Used by

Dependency tree · two levels

8 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, Corollary 16.12 (standard reference, not scraped)