PropositionStatement: Literature-sourcedProof: AI-adaptedSession-authored (Fable 5 assisted)precheck passaudited 2026-08-27

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A unique abelian minimal normal subgroup gives affine type

Statement

Let $G \leq Sym (Ω)$ be a finite faithful primitive group, and suppose that $V ⊴ G$ is its unique minimal normal subgroup and that $V$ is abelian. Then:

$V$ is regular on $Ω$ ;
$V$ is elementary abelian, so $V ≅ (F_{p})^{d}$ for some prime $p$ ;
for every $α \in Ω$ , the point stabilizer $G_{α}$ acts faithfully and irreducibly on the vector space $V$ .

In the O'Nan-Scott language, $G$ is of affine type.

Facts & Assumptions

Given: A finite faithful primitive group $G \leq Sym (Ω)$ with unique abelian minimal normal subgroup $V$ .

[L1]

Every nontrivial abelian normal subgroup of a faithful primitive action is regular (Abelian normal subgroups of faithful primitive actions are regular).

[L2]

Every minimal normal subgroup of a finite group is characteristically simple (Minimal normal subgroups of finite groups are characteristically simple).

[A1]

A finite abelian characteristically simple group is elementary abelian.

[L3]

An elementary abelian $p$ -group is canonically a vector space over $F_{p}$ (An elementary abelian $p$ -group has a canonical $F_{p}$ -vector-space structure).

[L4]

Every finite elementary abelian $p$ -group has a finite basis over $F_{p}$ (Finite elementary abelian $p$ -groups have bases, basis extension, and a well-defined dimension).

Proof

technique · direct

1.1

By [L1], the abelian normal subgroup $V$ is regular on $Ω$ .

givenL1

1.2

By [L2], the minimal normal subgroup $V$ is characteristically simple; as it is also abelian, [A1] shows that $V$ is elementary abelian. Facts [L3] and [L4] therefore identify $V$ with $(F_{p})^{d}$ for some prime $p$ and some $d \geq 1$ .

L2L3L4A1

2.1

Fix $α \in Ω$ . Because $V$ is regular, every $g \in G_{α}$ acts on $V$ by conjugation and the kernel of this action is $G_{α} \cap C_{G} (V)$ . If a nontrivial element of $G_{α}$ centralized $V$ , then it would fix every point $v α$ with $v \in V$ , contradicting faithfulness; so the action is faithful. If $W < V$ were a nontrivial proper $G_{α}$ -invariant subgroup, then $W$ would be normal in $V G_{α} = G$ , contradicting minimality of $V$ . Thus the action is irreducible, and $G$ is of affine type.

step 1.1step 1.2algebra∎

Depends on

Used by

Dependency tree · two levels

25 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

J. S. Milne, Group Theory, Exercise 6.31 (standard reference, not scraped)
Leonard H. Soicher, Primitive permutation groups (standard reference, not scraped)