Alphabeta Math
False statementConstruction: Literature-sourcedVerification: AI-generatedSession-authored (Fable 5 assisted)precheck passjudge pass (gpt-5.6-terra)audited 2026-08-26
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FALSE: every transitive action is primitive

Statement

Every transitive action is primitive.

Facts & Assumptions

Given: The regular action of Cn on itself for a composite integer n.

[L1]

In the regular cyclic action, the blocks are exactly the cosets of subgroups. For composite n this yields nontrivial blocks (Blocks in a regular cyclic action are cosets of subgroups).

Refutation

technique · direct
1.1

The regular action of Cn on itself is transitive, since every point is reached by a translation.

givenalgebra
1.2

When n is composite, [L1] gives a nontrivial block system coming from a proper nontrivial subgroup of Cn. So this transitive action is not primitive.

L1
2.1

Steps 1.1 and 1.2 refute the statement.

step 1.1step 1.2∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

3 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