LemmaStatement: Literature-sourcedProof: AI-generatedprecheck 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.

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Two distinct minimal normal subgroups of a primitive group are regular

Statement

Let $G \leq Sym (Ω)$ be finite, faithful, and primitive, and let $M, N ⊴ G$ be distinct minimal normal subgroups. Then both $M$ and $N$ act regularly on $Ω$ .

Facts & Assumptions

Given: A finite faithful primitive action of $G$ on $Ω$ and distinct minimal normal subgroups $M, N ⊴ G$ .

[L1]

Distinct minimal normal subgroups centralize one another (Distinct minimal normal subgroups centralize one another).

[L2]

Every minimal normal subgroup of a finite faithful primitive group is transitive (Minimal normal subgroups of faithful primitive groups are transitive).

Proof

technique · direct

1.1L1L2

By [L2], both $M$ and $N$ are transitive on $Ω$ . By [L1], every element of $M$ commutes with every element of $N$ .

2.1step 1.1choosealgebra

Fix $α \in Ω$ , and let $m \in M_{α}$ . For any $β \in Ω$ , choose $n \in N$ with $n α = β$ ; then $m β = m n α = n m α = n α = β$ . Hence every element of $M_{α}$ fixes every point of $Ω$ , so faithfulness gives $M_{α} = 1$ .

3.1step 2.1∎

The subgroup $M$ is transitive with trivial point stabilizer, so it is regular. By symmetry the same argument applies to $N$ .

Depends on

Used by

A finite primitive group has at most two minimal normal subgroups Corollary
A primitive group with two regular minimal normal subgroups Example
FALSE: every primitive group has a unique minimal normal subgroup False statement
Finite 2-transitive groups have affine or almost simple socle type Proposition

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

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