ExampleConstruction: Literature-sourcedVerification: AI-generatedSession-authored (Fable 5 assisted)precheck passaudited 2026-08-27

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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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The natural action of AGL(1,p) is affine type

Example

Let $p$ be prime. The natural action of $AGL (1, p) = F_{p} ⋊ F_{p}^{\times}$ on the set $F_{p}$ is of affine type.

Facts & Assumptions

Given: The semidirect product action $(b, a) \cdot x = a x + b$ of $AGL (1, p)$ on $F_{p}$ .

[L1]

A faithful primitive group with a unique abelian minimal normal subgroup is of affine type (A unique abelian minimal normal subgroup gives affine type).

[A1]

The translation subgroup $F_{p} \times {1}$ is normal, abelian, and regular on $F_{p}$ .

Verification

technique · direct

1.1

The translation subgroup $V = F_{p} \times {1}$ is elementary abelian of order $p$ and acts regularly by $x \mapsto x + b$ .

givenA1

2.1

Let $N ⊴ AGL (1, p)$ be nontrivial. If $N \cap V \neq 1$ , then $V \leq N$ because $V$ has prime order. If $N \cap V = 1$ , then the image of $N$ in the quotient $AGL (1, p) / V ≅ F_{p}^{\times}$ is nontrivial; choose $(c, a) \in N$ with $a \neq 1$ . For any $b \in F_{p}$ , the commutator $(b, 1) (c, a) (- b, 1) (c, a)^{- 1} = ((1 - a) b, 1)$ lies in $N \cap V$ , contradiction. So every nontrivial normal subgroup contains $V$ , and $V$ is the unique minimal normal subgroup.

step 1.1choosealgebra

3.1

Therefore [L1] applies, and the natural action of $AGL (1, p)$ is of affine type.

L1step 2.1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

11 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)