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

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The socle of a finite solvable primitive group is elementary abelian and regular

Example

If $G \leq Sym (Ω)$ is finite, solvable, faithful, primitive, and of degree at least $2$ , then its socle is the unique regular elementary abelian minimal normal subgroup.

Facts & Assumptions

Given: A finite solvable faithful primitive permutation group $G \leq Sym (Ω)$ of degree at least $2$ .

[A1]

In a finite solvable primitive group of degree at least $2$ , a minimal normal subgroup is elementary abelian and regular.

[A2]

In a finite solvable primitive group of degree at least $2$ , that minimal normal subgroup is unique.

Verification

technique · direct

1.1

Let $N$ be a minimal normal subgroup of $G$ . The primitive-solvable fact [A1] shows that $N$ is elementary abelian and regular.

givenA1

2.1

The uniqueness statement [A2] says that this minimal normal subgroup is the only one.

A2step 1.1

3.1

The socle is generated by the minimal normal subgroups, so with uniqueness from step 2.1 it equals $N$ . Therefore the socle of $G$ is the unique regular elementary abelian minimal normal subgroup.

step 1.1step 2.1∎

Depends on

A unique abelian minimal normal subgroup gives affine type

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

J. S. Milne, Group Theory, Exercise 6.31 (standard reference, not scraped)