CounterexampleConstruction: Literature-sourcedVerification: AI-adaptedSession-authored (Fable 5 assisted)precheck passjudge pass (gpt-5.6-terra)audited 2026-08-27

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A transitive imprimitive action can have a nontransitive minimal normal subgroup

Statement refuted

The theorem Minimal normal subgroups of faithful primitive groups are transitive requires primitivity. Mere transitivity does not force a minimal normal subgroup to be transitive.

Facts & Assumptions

Given: A nonabelian finite simple group $T$ and the action of $(T \times T) ⋊ C_{2}$ on the disjoint union of two copies of $T$ , where $T \times T$ acts by left regular action on each copy and $C_{2}$ swaps the two copies.

[A1]

This action is transitive because the swapping involution exchanges the two blocks, but it is imprimitive because the two copies of $T$ form a nontrivial block system.

[A2]

The subgroup $T \times T$ is a minimal normal subgroup of the full semidirect product, and it preserves each block setwise.

Counterexample

technique · direct

1.1

By [A1], the action is transitive and imprimitive.

givenA1

1.2

By [A2], the subgroup $N = T \times T$ is minimal normal, but because $N$ preserves each copy of $T$ setwise, it has two orbits and is therefore not transitive.

2.1

Hence a transitive action can have a nontransitive minimal normal subgroup once primitivity is dropped.

step 1.1step 1.2∎

Depends on

Used by

Nothing in the library uses this result yet.

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

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