False statementConstruction: Literature-sourcedVerification: AI-generatedSession-authored (Fable 5 assisted)precheck passaudited 2026-08-26

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FALSE: transitivity alone forces nontrivial normal subgroups to be transitive

Statement

In every faithful transitive action, each nontrivial normal subgroup acts transitively.

Facts & Assumptions

Given: The faithful regular action of $C_{4}$ on itself.

[L1]

In a faithful primitive action, every nontrivial normal subgroup is transitive (Iwasawa's simplicity criterion for primitive actions).

[L2]

In the regular cyclic action of composite degree, proper nontrivial subgroups yield nontrivial blocks (Blocks in a regular cyclic action are cosets of subgroups).

Refutation

technique · direct

1.1

The action of $C_{4}$ on itself is transitive, but [L2] shows it is not primitive.

2.1

The subgroup $2 C_{4} = {0, 2}$ is nontrivial, normal, and not transitive: its orbits are ${0, 2}$ and ${1, 3}$ . So the transitivity conclusion singled out in [L1] fails once primitivity is removed.

step 1.1algebra

3.1

The regular action is faithful and transitive, while step 2.1 gives a nontrivial normal subgroup that is not transitive. Therefore the statement is false.

step 1.1step 2.1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

7 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

K. Conrad, Transitive Group Actions (standard reference, not scraped)
P. J. Cameron, Classical Groups, Sections 2.3-2.4 (standard reference, not scraped)