TheoremStatement: 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 faithful primitive groups are transitive

Statement

Let $G \leq Sym (Ω)$ be finite, faithful, and primitive, and let $N ⊴ G$ be a minimal normal subgroup. Then $N$ acts transitively on $Ω$ .

Facts & Assumptions

Given: A finite faithful primitive action of $G$ on $Ω$ and a minimal normal subgroup $N ⊴ G$ .

[L1]

In a primitive action, every normal subgroup is either transitive or contained in the kernel (Normal subgroups of a primitive action are transitive or lie in the kernel).

[A1]

A faithful action has trivial kernel.

Proof

technique · direct

1.1

By [L1], the normal subgroup $N$ is either transitive or contained in the kernel of the action.

givenL1

2.1

The action is faithful, so [A1] gives trivial kernel. Because $N$ is a minimal normal subgroup, it is nontrivial, so the kernel-contained alternative from step 1.1 is impossible. Hence $N$ is transitive.

A1step 1.1∎

Depends on

Used by

Dependency tree · two levels

6 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 4 (standard reference, not scraped)
Leonard H. Soicher, Primitive permutation groups (standard reference, not scraped)